id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-12900 | Element of Pascal's Triangle is Sum of Diagonal or Column starting above it going Upwards | Consider Pascal's triangle:
:{{Definition:Pascal's Triangle}}
Let $\tuple {n, m}$ be the element in the $n$th row and $m$th column.
Then:
:$\tuple {n, m} = \ds \sum_{k \mathop \ge 0} \tuple {n - k - 1, m - 1}$
and:
:$\tuple {n, m} = \ds \sum_{k \mathop \ge 0} \tuple {n - k - 1, m - k - 1}$ | We have $\tuple {n, m} = \dbinom n m$, and we have:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop \ge 0} \binom {n - k - 1} {m - 1}
| r = \sum_{k \mathop = 0}^{n - m} \binom {m - 1 + k} {m - 1}
}}
{{eqn | r = \binom {m - 1 + n - m + 1} {m - 1 + 1}
| c = Rising Sum of Binomial Coefficients
}}
{{eqn | r = \binom ... | Consider [[Definition:Pascal's Triangle|Pascal's triangle]]:
:{{Definition:Pascal's Triangle}}
Let $\tuple {n, m}$ be the [[Definition:Element of Array|element]] in the $n$th [[Definition:Row of Pascal's Triangle|row]] and $m$th [[Definition:Column of Pascal's Triangle|column]].
Then:
:$\tuple {n, m} = \ds \sum_{k \m... | We have $\tuple {n, m} = \dbinom n m$, and we have:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop \ge 0} \binom {n - k - 1} {m - 1}
| r = \sum_{k \mathop = 0}^{n - m} \binom {m - 1 + k} {m - 1}
}}
{{eqn | r = \binom {m - 1 + n - m + 1} {m - 1 + 1}
| c = [[Rising Sum of Binomial Coefficients]]
}}
{{eqn | r = \b... | Element of Pascal's Triangle is Sum of Diagonal or Column starting above it going Upwards | https://proofwiki.org/wiki/Element_of_Pascal's_Triangle_is_Sum_of_Diagonal_or_Column_starting_above_it_going_Upwards | https://proofwiki.org/wiki/Element_of_Pascal's_Triangle_is_Sum_of_Diagonal_or_Column_starting_above_it_going_Upwards | [
"Pascal's Triangle"
] | [
"Definition:Pascal's Triangle",
"Definition:Array/Element",
"Definition:Pascal's Triangle/Row",
"Definition:Pascal's Triangle/Column"
] | [
"Rising Sum of Binomial Coefficients",
"Symmetry Rule for Binomial Coefficients",
"Rising Sum of Binomial Coefficients"
] |
proofwiki-12901 | Power of Moved Element is Moved | Let $S_n$ denote the symmetric group on $n$ letters.
Let $\sigma \in S_n$.
Then for all $m \in \Z$:
:$i \notin \Fix \sigma \implies \map {\sigma^m} i \notin \Fix \sigma$
where $\Fix \sigma$ denotes the set of fixed elements of $\sigma$. | {{AimForCont}} that there exists some $i \notin \Fix \sigma$ and some $m \in \Z$ such that $\map {\sigma^m} i \in \Fix \sigma$.
Then:
{{begin-eqn}}
{{eqn | l = \map {\paren {\sigma \circ \sigma^m} } i
| r = \map {\sigma^{m + 1} } i
}}
{{eqn | r = \map {\sigma^m} i
| c = {{Defof|Fixed Element under Permutati... | Let $S_n$ denote the [[Definition:Symmetric Group on n Letters|symmetric group on $n$ letters]].
Let $\sigma \in S_n$.
Then for all $m \in \Z$:
:$i \notin \Fix \sigma \implies \map {\sigma^m} i \notin \Fix \sigma$
where $\Fix \sigma$ denotes the [[Definition:Set of Fixed Elements|set of fixed elements]] of $\sigma... | {{AimForCont}} that there exists some $i \notin \Fix \sigma$ and some $m \in \Z$ such that $\map {\sigma^m} i \in \Fix \sigma$.
Then:
{{begin-eqn}}
{{eqn | l = \map {\paren {\sigma \circ \sigma^m} } i
| r = \map {\sigma^{m + 1} } i
}}
{{eqn | r = \map {\sigma^m} i
| c = {{Defof|Fixed Element under Permutat... | Power of Moved Element is Moved | https://proofwiki.org/wiki/Power_of_Moved_Element_is_Moved | https://proofwiki.org/wiki/Power_of_Moved_Element_is_Moved | [
"Symmetric Groups"
] | [
"Definition:Symmetric Group/n Letters",
"Definition:Fixed Element under Permutation/Set of Fixed Elements"
] | [
"Definition:Fixed Element under Permutation/Moved",
"Definition:Contradiction",
"Category:Symmetric Groups"
] |
proofwiki-12902 | Triangular Numbers in Geometric Sequence | The numbers:
:$1, 6, 36$
are the smallest triangular numbers in geometric sequence. | {{begin-eqn}}
{{eqn | l = 6 \div 1
| r = 6
| c =
}}
{{eqn | l = 36 \div 6
| r = 6
| c =
}}
{{end-eqn}}
Hence the common ratio is $6$.
{{qed}} | The numbers:
:$1, 6, 36$
are the smallest [[Definition:Triangular Number|triangular numbers]] in [[Definition:Geometric Sequence|geometric sequence]]. | {{begin-eqn}}
{{eqn | l = 6 \div 1
| r = 6
| c =
}}
{{eqn | l = 36 \div 6
| r = 6
| c =
}}
{{end-eqn}}
Hence the [[Definition:Common Ratio|common ratio]] is $6$.
{{qed}} | Triangular Numbers in Geometric Sequence | https://proofwiki.org/wiki/Triangular_Numbers_in_Geometric_Sequence | https://proofwiki.org/wiki/Triangular_Numbers_in_Geometric_Sequence | [
"Triangular Numbers",
"Geometric Sequences"
] | [
"Definition:Triangular Number",
"Definition:Geometric Sequence"
] | [
"Definition:Geometric Sequence/Common Ratio"
] |
proofwiki-12903 | Square Numbers which are Divisor Sum values | The sequence of square numbers which are the divisor sum value of a (strictly) positive integer begins:
:$1, 4, 36, 121, 144, 256, 324, 400, 576, 784, 900, 961, \ldots$
{{OEIS|A038688}} | {{begin-eqn}}
{{eqn | l = 1
| r = \map {\sigma_1} 1
| c = {{DSFLink|1}}
}}
{{eqn | l = 4
| r = \map {\sigma_1} 3
| c = {{DSFLink|3}}
}}
{{eqn | l = 36
| r = \map {\sigma_1} {22}
| c = {{DSFLink|22}}
}}
{{eqn | l = 121
| r = \map {\sigma_1} {81}
| c = {{DSFLink|81}}
}}
{{e... | The [[Definition:Integer Sequence|sequence]] of [[Definition:Square Number|square numbers]] which are the [[Definition:Divisor Sum Function|divisor sum value]] of a [[Definition:Strictly Positive Integer|(strictly) positive integer]] begins:
:$1, 4, 36, 121, 144, 256, 324, 400, 576, 784, 900, 961, \ldots$
{{OEIS|A0386... | {{begin-eqn}}
{{eqn | l = 1
| r = \map {\sigma_1} 1
| c = {{DSFLink|1}}
}}
{{eqn | l = 4
| r = \map {\sigma_1} 3
| c = {{DSFLink|3}}
}}
{{eqn | l = 36
| r = \map {\sigma_1} {22}
| c = {{DSFLink|22}}
}}
{{eqn | l = 121
| r = \map {\sigma_1} {81}
| c = {{DSFLink|81}}
}}
{{e... | Square Numbers which are Divisor Sum values | https://proofwiki.org/wiki/Square_Numbers_which_are_Divisor_Sum_values | https://proofwiki.org/wiki/Square_Numbers_which_are_Divisor_Sum_values | [
"Divisor Sum Function",
"Square Numbers"
] | [
"Definition:Integer Sequence",
"Definition:Square Number",
"Definition:Divisor Sum Function",
"Definition:Strictly Positive/Integer"
] | [] |
proofwiki-12904 | Join and Meet in Inclusion Ordered Set of Topology | Let $T = \left({S, \tau}\right)$ be a topological space.
Let $L = \left({\tau, \preceq}\right)$ be an inclusion ordered set of $\tau$.
Let $X, Y \in \tau$.
Then $X \vee Y = X \cup Y$ and $X \wedge Y = X \cap Y$ | By definition of topological space:
:$X \cup Y, X \cap Y \in \tau$
Thus by Join in Inclusion Ordered Set ans Meet in Inclusion Ordered Set:
:the result holds.
{{qed}} | Let $T = \left({S, \tau}\right)$ be a [[Definition:Topological Space|topological space]].
Let $L = \left({\tau, \preceq}\right)$ be an [[Definition:Inclusion Ordered Set|inclusion ordered set]] of $\tau$.
Let $X, Y \in \tau$.
Then $X \vee Y = X \cup Y$ and $X \wedge Y = X \cap Y$ | By definition of [[Definition:Topological Space|topological space]]:
:$X \cup Y, X \cap Y \in \tau$
Thus by [[Join in Inclusion Ordered Set]] ans [[Meet in Inclusion Ordered Set]]:
:the result holds.
{{qed}} | Join and Meet in Inclusion Ordered Set of Topology | https://proofwiki.org/wiki/Join_and_Meet_in_Inclusion_Ordered_Set_of_Topology | https://proofwiki.org/wiki/Join_and_Meet_in_Inclusion_Ordered_Set_of_Topology | [
"Join and Meet",
"Topology"
] | [
"Definition:Topological Space",
"Definition:Inclusion Ordered Set"
] | [
"Definition:Topological Space",
"Join in Inclusion Ordered Set",
"Meet in Inclusion Ordered Set"
] |
proofwiki-12905 | Join in Inclusion Ordered Set | Let $P = \left({X, \subseteq}\right)$ be an inclusion ordered set.
Let $A, B \in X$ such that
:$A \cup B \in X$
Then $A \vee B = A \cup B$ | By Set is Subset of Union:
:$A \subseteq A \cup B$ and $B \subseteq A \cup B$
By definition:
:$A \cup B$ is upper bound for $\left\{ {A, B}\right\}$
We will prove that
:$\forall C \in X: C$ is upper bound for $\left\{ {A, B}\right\} \implies A \cup B \subseteq C$
Let $C \in X$ such that
:$C$ is upper bound for $\left\{... | Let $P = \left({X, \subseteq}\right)$ be an [[Definition:Subset|inclusion]] [[Definition:Ordered Set|ordered set]].
Let $A, B \in X$ such that
:$A \cup B \in X$
Then $A \vee B = A \cup B$ | By [[Set is Subset of Union]]:
:$A \subseteq A \cup B$ and $B \subseteq A \cup B$
By definition:
:$A \cup B$ is [[Definition:Upper Bound of Set|upper bound]] for $\left\{ {A, B}\right\}$
We will prove that
:$\forall C \in X: C$ is [[Definition:Upper Bound of Set|upper bound]] for $\left\{ {A, B}\right\} \implies A \c... | Join in Inclusion Ordered Set | https://proofwiki.org/wiki/Join_in_Inclusion_Ordered_Set | https://proofwiki.org/wiki/Join_in_Inclusion_Ordered_Set | [
"Join and Meet"
] | [
"Definition:Subset",
"Definition:Ordered Set"
] | [
"Set is Subset of Union",
"Definition:Upper Bound of Set",
"Definition:Upper Bound of Set",
"Definition:Upper Bound of Set",
"Definition:Upper Bound of Set",
"Union is Smallest Superset",
"Definition:Supremum of Set",
"Definition:Join (Order Theory)"
] |
proofwiki-12906 | Characterization of Prime Element in Inclusion Ordered Set of Topology | Let $T = \struct {S, \tau}$ be a topological space.
Let $L = \struct {\tau, \preceq}$ be an inclusion ordered set of $\tau$.
Let $Z \in \tau$.
Then $Z$ is prime element in $L$ {{iff}}:
:$\forall X, Y \in \tau: X \cap Y \subseteq Z \implies X \subseteq Z \lor Y \subseteq Z$ | === Sufficient Condition ===
Assume that
:$Z$ is prime element in $L$.
Let $X, T \in \tau$ such that:
:$X \cap Y \subseteq Z$
By Join and Meet in Inclusion Ordered Set of Topology and definition of inclusion ordered set:
:$X \wedge Y \preceq Z$
By definition of prime element:
:$X \preceq Z$ or $Y \preceq Z$
Thus by def... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $L = \struct {\tau, \preceq}$ be an [[Definition:Inclusion Ordered Set|inclusion ordered set]] of $\tau$.
Let $Z \in \tau$.
Then $Z$ is [[Definition:Prime Element (Order Theory)|prime element]] in $L$ {{iff}}:
:$\forall X, Y \... | === Sufficient Condition ===
Assume that
:$Z$ is [[Definition:Prime Element (Order Theory)|prime element]] in $L$.
Let $X, T \in \tau$ such that:
:$X \cap Y \subseteq Z$
By [[Join and Meet in Inclusion Ordered Set of Topology]] and definition of [[Definition:Inclusion Ordered Set|inclusion ordered set]]:
:$X \wedge ... | Characterization of Prime Element in Inclusion Ordered Set of Topology | https://proofwiki.org/wiki/Characterization_of_Prime_Element_in_Inclusion_Ordered_Set_of_Topology | https://proofwiki.org/wiki/Characterization_of_Prime_Element_in_Inclusion_Ordered_Set_of_Topology | [
"Prime Elements",
"Topology"
] | [
"Definition:Topological Space",
"Definition:Inclusion Ordered Set",
"Definition:Prime Element (Order Theory)"
] | [
"Definition:Prime Element (Order Theory)",
"Join and Meet in Inclusion Ordered Set of Topology",
"Definition:Inclusion Ordered Set",
"Definition:Prime Element (Order Theory)",
"Definition:Inclusion Ordered Set",
"Join and Meet in Inclusion Ordered Set of Topology",
"Definition:Inclusion Ordered Set",
... |
proofwiki-12907 | 2-Digit Numbers divisible by both Product and Sum of Digits | The $2$-digit positive integers which are divisible by both the sum and product of their digits are:
:$12, 24, 36$ | We have:
{{begin-eqn}}
{{eqn | l = 12
| r = 4 \times \paren {1 + 2}
| c =
}}
{{eqn | r = 6 \times \paren {1 \times 2}
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 24
| r = 4 \times \paren {2 + 4}
| c =
}}
{{eqn | r = 3 \times \paren {2 \times 4}
| c =
}}
{{end-eqn}}
{{begin-eqn... | The [[Definition:Digit|$2$-digit]] [[Definition:Positive Integer|positive integers]] which are [[Definition:Divisor of Integer|divisible]] by both the [[Definition:Integer Addition|sum]] and [[Definition:Integer Multiplication|product]] of their [[Definition:Digit|digits]] are:
:$12, 24, 36$ | We have:
{{begin-eqn}}
{{eqn | l = 12
| r = 4 \times \paren {1 + 2}
| c =
}}
{{eqn | r = 6 \times \paren {1 \times 2}
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 24
| r = 4 \times \paren {2 + 4}
| c =
}}
{{eqn | r = 3 \times \paren {2 \times 4}
| c =
}}
{{end-eqn}}
{{begin-... | 2-Digit Numbers divisible by both Product and Sum of Digits | https://proofwiki.org/wiki/2-Digit_Numbers_divisible_by_both_Product_and_Sum_of_Digits | https://proofwiki.org/wiki/2-Digit_Numbers_divisible_by_both_Product_and_Sum_of_Digits | [
"Recreational Mathematics"
] | [
"Definition:Digit",
"Definition:Positive/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Addition/Integers",
"Definition:Multiplication/Integers",
"Definition:Digit"
] | [
"Definition:Digit",
"Definition:Positive/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Addition/Integers",
"Definition:Multiplication/Integers",
"Definition:Digit",
"Definition:Digit",
"Definition:Multiplication/Integers",
"Definition:Digit",
"Definition:Positive/Integer",
"Defin... |
proofwiki-12908 | 37 is Second Number whose Period of Reciprocal is 3 | $37$ is the $2$nd positive integer (after $27$) the decimal expansion of whose reciprocal has a period of $3$:
:$\dfrac 1 {37} = 0 \cdotp \dot 02 \dot 7$ | From Reciprocal of $37$:
{{:Reciprocal of 37}}
Counting the digits, it is seen that this has a period of recurrence of $3$.
It can be determined by inspection of all smaller integers that this is indeed the $2$nd to have a period of $3$.
{{qed}} | $37$ is the $2$nd [[Definition:Positive Integer|positive integer]] (after $27$) the [[Definition:Decimal Expansion|decimal expansion]] of whose [[Definition:Reciprocal|reciprocal]] has a [[Definition:Period of Recurrence|period]] of $3$:
:$\dfrac 1 {37} = 0 \cdotp \dot 02 \dot 7$ | From [[Reciprocal of 37|Reciprocal of $37$]]:
{{:Reciprocal of 37}}
Counting the digits, it is seen that this has a [[Definition:Period of Recurrence|period of recurrence]] of $3$.
It can be determined by inspection of all smaller integers that this is indeed the $2$nd to have a [[Definition:Period of Recurrence|perio... | 37 is Second Number whose Period of Reciprocal is 3 | https://proofwiki.org/wiki/37_is_Second_Number_whose_Period_of_Reciprocal_is_3 | https://proofwiki.org/wiki/37_is_Second_Number_whose_Period_of_Reciprocal_is_3 | [
"37",
"Examples of Reciprocals"
] | [
"Definition:Positive/Integer",
"Definition:Decimal Expansion",
"Definition:Reciprocal",
"Definition:Basis Expansion/Recurrence/Period"
] | [
"Reciprocal of 37",
"Definition:Basis Expansion/Recurrence/Period",
"Definition:Basis Expansion/Recurrence/Period"
] |
proofwiki-12909 | Centered Hexagonal Number as Sum of Triangular Numbers | Let $C_n$ be the $n$th centered hexagonal number.
Then:
:$C_n = 6 T_{n - 1} + 1$
where $T_{n - 1}$ denotes the $n - 1$th triangular number. | {{begin-eqn}}
{{eqn | l = C_n
| r = 3 n \paren {n - 1} + 1
| c = Closed Form for Centered Hexagonal Numbers
}}
{{eqn | r = 6 \paren {\dfrac {\paren {n - 1} n} 2} + 1
| c =
}}
{{eqn | r = 6 T_{n - 1} + 1
| c = Closed Form for Triangular Numbers
}}
{{end-eqn}}
{{qed}} | Let $C_n$ be the $n$th [[Definition:Centered Hexagonal Number|centered hexagonal number]].
Then:
:$C_n = 6 T_{n - 1} + 1$
where $T_{n - 1}$ denotes the $n - 1$th [[Definition:Triangular Number|triangular number]]. | {{begin-eqn}}
{{eqn | l = C_n
| r = 3 n \paren {n - 1} + 1
| c = [[Closed Form for Centered Hexagonal Numbers]]
}}
{{eqn | r = 6 \paren {\dfrac {\paren {n - 1} n} 2} + 1
| c =
}}
{{eqn | r = 6 T_{n - 1} + 1
| c = [[Closed Form for Triangular Numbers]]
}}
{{end-eqn}}
{{qed}} | Centered Hexagonal Number as Sum of Triangular Numbers | https://proofwiki.org/wiki/Centered_Hexagonal_Number_as_Sum_of_Triangular_Numbers | https://proofwiki.org/wiki/Centered_Hexagonal_Number_as_Sum_of_Triangular_Numbers | [
"Centered Hexagonal Number as Sum of Triangular Numbers",
"Centered Hexagonal Numbers",
"Triangular Numbers"
] | [
"Definition:Centered Hexagonal Number",
"Definition:Triangular Number"
] | [
"Closed Form for Centered Hexagonal Numbers",
"Closed Form for Triangular Numbers"
] |
proofwiki-12910 | Closed Form for Centered Hexagonal Numbers | Let $C_n$ be the $n$th centered hexagonal number.
Then:
:$C_n = 3 n \paren {n - 1} + 1$ | By the definition of centered hexagonal number:
{{begin-eqn}}
{{eqn | l = C_n
| r = 1 + \sum_{k \mathop = 1}^{n - 1} 6 k
| c = {{Defof|Centered Hexagonal Number}}
}}
{{eqn | r = 1 + 6 \sum_{k \mathop = 1}^{n - 1} k
| c =
}}
{{eqn | r = 6 \paren {\dfrac {n \paren {n - 1} } 2} + 1
| c = Closed Fo... | Let $C_n$ be the $n$th [[Definition:Centered Hexagonal Number|centered hexagonal number]].
Then:
:$C_n = 3 n \paren {n - 1} + 1$ | By the definition of [[Definition:Centered Hexagonal Number|centered hexagonal number]]:
{{begin-eqn}}
{{eqn | l = C_n
| r = 1 + \sum_{k \mathop = 1}^{n - 1} 6 k
| c = {{Defof|Centered Hexagonal Number}}
}}
{{eqn | r = 1 + 6 \sum_{k \mathop = 1}^{n - 1} k
| c =
}}
{{eqn | r = 6 \paren {\dfrac {n \pa... | Closed Form for Centered Hexagonal Numbers | https://proofwiki.org/wiki/Closed_Form_for_Centered_Hexagonal_Numbers | https://proofwiki.org/wiki/Closed_Form_for_Centered_Hexagonal_Numbers | [
"Centered Hexagonal Numbers"
] | [
"Definition:Centered Hexagonal Number"
] | [
"Definition:Centered Hexagonal Number",
"Closed Form for Triangular Numbers"
] |
proofwiki-12911 | Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers | The positive even integers which cannot be expressed as the sum of $2$ composite odd numbers are:
:$2, 4, 6, 8, 10, 12, 14, 16, 20, 22, 26, 28, 32, 38$
{{OEIS|A118081}} | The smallest composite odd numbers are $9$ and $15$, so trivially $2$ to $16$ and $20$ to $22$ cannot be expressed as the sum of $2$ composite odd numbers.
We have:
{{begin-eqn}}
{{eqn | l = 18
| r = 9 + 9
}}
{{eqn | l = 24
| r = 9 + 15
}}
{{eqn | l = 30
| r = 21 + 9
}}
{{eqn | r = 15 + 15
}}
{{eqn | ... | The [[Definition:Positive Integer|positive]] [[Definition:Even Integer|even integers]] which cannot be expressed as the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Composite Number|composite]] [[Definition:Odd Integer|odd numbers]] are:
:$2, 4, 6, 8, 10, 12, 14, 16, 20, 22, 26, 28, 32, 38$
{{OEIS|A118081}} | The smallest [[Definition:Composite Number|composite]] [[Definition:Odd Number|odd numbers]] are $9$ and $15$, so trivially $2$ to $16$ and $20$ to $22$ cannot be expressed as the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Composite Number|composite]] [[Definition:Odd Integer|odd numbers]].
We have:
{{beg... | Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers | https://proofwiki.org/wiki/Positive_Even_Integers_not_Expressible_as_Sum_of_2_Composite_Odd_Numbers | https://proofwiki.org/wiki/Positive_Even_Integers_not_Expressible_as_Sum_of_2_Composite_Odd_Numbers | [
"Composite Numbers",
"Odd Integers"
] | [
"Definition:Positive/Integer",
"Definition:Even Integer",
"Definition:Addition/Integers",
"Definition:Composite Number",
"Definition:Odd Integer"
] | [
"Definition:Composite Number",
"Definition:Odd Integer",
"Definition:Addition/Integers",
"Definition:Composite Number",
"Definition:Odd Integer",
"Definition:Composite Number",
"Definition:Odd Integer",
"Definition:Composite Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definit... |
proofwiki-12912 | Upper Closure is Compact in Topological Lattice | Let $L = \struct {S, \preceq, \tau}$ be a topological lattice.
Suppose that:
:for every subset $X$ of $S$ if $X$ is open, then $X$ is upper.
Let $x \in S$.
Then $x^\succeq$ is compact
where $x^\succeq$ denotes the upper closure of $x$. | Let $\FF$ be a set of subsets of $S$ such that:
:$\FF$ is open cover of $x^\succeq$
By definition of cover:
:$x^\succeq \subseteq \bigcup \FF$
By definitions of upper closure of element and reflexivity:
:$x \in x^\succeq$
By definition of subset:
:$x \in \bigcup \FF$
By definition of union:
:$\exists Y \in \FF: x \in Y... | Let $L = \struct {S, \preceq, \tau}$ be a [[Definition:Topological Lattice|topological lattice]].
Suppose that:
:for every [[Definition:Subset|subset]] $X$ of $S$ if $X$ is [[Definition:Open Set (Topology)|open]], then $X$ is [[Definition:Upper Section|upper]].
Let $x \in S$.
Then $x^\succeq$ is [[Definition:Compac... | Let $\FF$ be a [[Definition:Set of Sets|set]] of [[Definition:Subset|subsets]] of $S$ such that:
:$\FF$ is [[Definition:Open Cover|open cover]] of $x^\succeq$
By definition of [[Definition:Cover of Set|cover]]:
:$x^\succeq \subseteq \bigcup \FF$
By definitions of [[Definition:Upper Closure of Element|upper closure of... | Upper Closure is Compact in Topological Lattice | https://proofwiki.org/wiki/Upper_Closure_is_Compact_in_Topological_Lattice | https://proofwiki.org/wiki/Upper_Closure_is_Compact_in_Topological_Lattice | [
"Topological Order Theory",
"Compact Topological Spaces"
] | [
"Definition:Topological Lattice",
"Definition:Subset",
"Definition:Open Set/Topology",
"Definition:Upper Section",
"Definition:Compact Topological Space/Subspace",
"Definition:Upper Closure/Element"
] | [
"Definition:Set of Sets",
"Definition:Subset",
"Definition:Open Cover",
"Definition:Cover of Set",
"Definition:Upper Closure/Element",
"Definition:Reflexivity",
"Definition:Subset",
"Definition:Set Union/Set of Sets",
"Definition:Open Cover",
"Definition:Open Set/Topology",
"Definition:Upper Clo... |
proofwiki-12913 | Strictly Positive Real Numbers are Closed under Multiplication | The set $\R_{>0}$ of strictly positive real numbers is closed under multiplication:
:$\forall a, b \in \R_{> 0}: a \times b \in \R_{> 0}$ | Let $b > 0$.
From {{Real-number-axiom|O2}}:
:$a > c \implies a \times b > c \times b$
Thus setting $c = 0$:
:$a > 0 \implies a \times b > 0 \times b$
But from Real Zero is Zero Element:
:$0 \times b = 0$
Hence the result:
:$a, b > 0 \implies a \times b > 0$
{{qed}} | The [[Definition:Set|set]] $\R_{>0}$ of [[Definition:Strictly Positive Real Number|strictly positive real numbers]] is [[Definition:Closed Algebraic Structure|closed]] under [[Definition:Multiplication|multiplication]]:
:$\forall a, b \in \R_{> 0}: a \times b \in \R_{> 0}$ | Let $b > 0$.
From {{Real-number-axiom|O2}}:
:$a > c \implies a \times b > c \times b$
Thus setting $c = 0$:
:$a > 0 \implies a \times b > 0 \times b$
But from [[Real Zero is Zero Element]]:
:$0 \times b = 0$
Hence the result:
:$a, b > 0 \implies a \times b > 0$
{{qed}} | Strictly Positive Real Numbers are Closed under Multiplication/Proof 2 | https://proofwiki.org/wiki/Strictly_Positive_Real_Numbers_are_Closed_under_Multiplication | https://proofwiki.org/wiki/Strictly_Positive_Real_Numbers_are_Closed_under_Multiplication/Proof_2 | [
"Real Multiplication",
"Strictly Positive Real Numbers are Closed under Multiplication"
] | [
"Definition:Set",
"Definition:Strictly Positive/Real Number",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Multiplication"
] | [
"Real Zero is Zero Element"
] |
proofwiki-12914 | Positive Power Function on Non-negative Reals is Strictly Increasing | Let $a \in \Q_{> 0}$ be a strictly positive rational number.
Let $f_a: \R_{\ge 0} \to \R$ be the real function defined as:
:$\map {f_a} x = x^a$
Then $f_a$ is strictly increasing. | By the power rule for derivatives:
:$\map {D_x} {x^a} = a x^{a - 1}$
By power of positive real number is positive, it is seen that:
:$x > 0 \implies x^{a - 1} > 0$
By Strictly Positive Real Numbers are Closed under Multiplication, it follows that $\map {D_x} {x^a} > 0$ for all $x \in \openint 0 {+\infty}$.
Hence by Der... | Let $a \in \Q_{> 0}$ be a [[Definition:Strictly Positive|strictly positive]] [[Definition:Rational Number|rational number]].
Let $f_a: \R_{\ge 0} \to \R$ be the [[Definition:Real Function|real function]] defined as:
:$\map {f_a} x = x^a$
Then $f_a$ is [[Definition:Strictly Increasing Real Function|strictly increasin... | By the [[Power Rule for Derivatives/Rational Index|power rule for derivatives]]:
:$\map {D_x} {x^a} = a x^{a - 1}$
By [[Power of Positive Real Number is Positive/Rational Number|power of positive real number is positive]], it is seen that:
:$x > 0 \implies x^{a - 1} > 0$
By [[Strictly Positive Real Numbers are Clos... | Positive Power Function on Non-negative Reals is Strictly Increasing | https://proofwiki.org/wiki/Positive_Power_Function_on_Non-negative_Reals_is_Strictly_Increasing | https://proofwiki.org/wiki/Positive_Power_Function_on_Non-negative_Reals_is_Strictly_Increasing | [
"Examples of Strictly Increasing Real Functions",
"Powers"
] | [
"Definition:Strictly Positive",
"Definition:Rational Number",
"Definition:Real Function",
"Definition:Strictly Increasing/Real Function"
] | [
"Power Rule for Derivatives/Rational Index",
"Power of Positive Real Number is Positive/Rational Number",
"Strictly Positive Real Numbers are Closed under Multiplication",
"Derivative of Monotone Function",
"Definition:Strictly Increasing/Real Function",
"Category:Examples of Strictly Increasing Real Func... |
proofwiki-12915 | Smallest Consecutive Even Numbers such that Added to Divisor Count are Equal | $30$ is the smallest positive even integer $n$ with the property:
{{begin-eqn}}
{{eqn | l = n + \map {\sigma_0} n
| r = m
| c =
}}
{{eqn | l = \paren {n + 2} + \map {\sigma_0} {n + 2}
| r = m
| c =
}}
{{eqn | l = \paren {n + 4} + \map {\sigma_0} {n + 4}
| r = m
| c =
}}
{{end-eqn}... | From Divisor Count Function from Prime Decomposition, we have:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where the prime decomposition of $n$ is:
:$n = p_1^{k_1} p_2^{k_2} \ldots p_r^{k_r}$
{{begin-eqn}}
{{eqn | l = 2 + \map {\sigma_0} 2
| r = 2 + 2
| rr= = 4
| c = as $2 = 2^... | $30$ is the smallest [[Definition:Positive Integer|positive]] [[Definition:Even Integer|even integer]] $n$ with the property:
{{begin-eqn}}
{{eqn | l = n + \map {\sigma_0} n
| r = m
| c =
}}
{{eqn | l = \paren {n + 2} + \map {\sigma_0} {n + 2}
| r = m
| c =
}}
{{eqn | l = \paren {n + 4} + \ma... | From [[Divisor Count Function from Prime Decomposition]], we have:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where the [[Definition:Prime Decomposition|prime decomposition]] of $n$ is:
:$n = p_1^{k_1} p_2^{k_2} \ldots p_r^{k_r}$
{{begin-eqn}}
{{eqn | l = 2 + \map {\sigma_0} 2
| r = 2 ... | Smallest Consecutive Even Numbers such that Added to Divisor Count are Equal | https://proofwiki.org/wiki/Smallest_Consecutive_Even_Numbers_such_that_Added_to_Divisor_Count_are_Equal | https://proofwiki.org/wiki/Smallest_Consecutive_Even_Numbers_such_that_Added_to_Divisor_Count_are_Equal | [
"Divisor Count Function",
"Even Integers"
] | [
"Definition:Positive/Integer",
"Definition:Even Integer",
"Definition:Positive/Integer",
"Definition:Divisor Count Function",
"Definition:Divisor (Algebra)/Integer"
] | [
"Divisor Count Function from Prime Decomposition",
"Definition:Prime Decomposition"
] |
proofwiki-12916 | 2-Digit Positive Integer equals Product plus Sum of Digits iff ends in 9 | Let $n$ be a $2$-digit positive integer.
Then:
:$n$ equals the sum added to the product of its digits
{{iff}}:
:the last digit of $n$ is $9$. | Let $n = 10 x + y$ where $0 < x \le 9, 0 \le y \le 9$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {x + y} + \paren {x y}
| r = 10 x + y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = x y - 9 x
| r = 0
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = x \paren {y - 9}
| r = 0
| c =
}}... | Let $n$ be a [[Definition:Digit|$2$-digit]] [[Definition:Positive Integer|positive integer]].
Then:
:$n$ equals the [[Definition:Integer Addition|sum]] added to the [[Definition:Integer Multiplication|product]] of its [[Definition:Digit|digits]]
{{iff}}:
:the last [[Definition:Digit|digit]] of $n$ is $9$. | Let $n = 10 x + y$ where $0 < x \le 9, 0 \le y \le 9$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {x + y} + \paren {x y}
| r = 10 x + y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = x y - 9 x
| r = 0
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = x \paren {y - 9}
| r = 0
| c =
}... | 2-Digit Positive Integer equals Product plus Sum of Digits iff ends in 9 | https://proofwiki.org/wiki/2-Digit_Positive_Integer_equals_Product_plus_Sum_of_Digits_iff_ends_in_9 | https://proofwiki.org/wiki/2-Digit_Positive_Integer_equals_Product_plus_Sum_of_Digits_iff_ends_in_9 | [
"Recreational Mathematics"
] | [
"Definition:Digit",
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Multiplication/Integers",
"Definition:Digit",
"Definition:Digit"
] | [] |
proofwiki-12917 | Infimum of Open Set is Way Below Element in Complete Scott Topological Lattice | Let $T = \struct {S, \preceq, \tau}$ be a complete topological lattice with Scott topology.
Let $X$ be an open subset of $S$,
Let $x \in X$.
Then $\inf X \ll x$
where $\ll$ denotes the way below relation. | By Open iff Upper and with Property (S) in Scott Topological Lattice:
:$X$ is upper and has property (S).
Let $D$ be a directed subset of $S$ such that
:$x \preceq \sup D$
By definition of upper section:
:$\sup D \in X$
By definition of property (S):
:$\exists y \in D: \forall d \in D: y \preceq d \implies d \in X$
By ... | Let $T = \struct {S, \preceq, \tau}$ be a [[Definition:Complete Lattice|complete]] [[Definition:Topological Lattice|topological lattice]] with [[Definition:Scott Topology|Scott topology]].
Let $X$ be an [[Definition:Open Set (Topology)|open]] [[Definition:Subset|subset]] of $S$,
Let $x \in X$.
Then $\inf X \ll x$
... | By [[Open iff Upper and with Property (S) in Scott Topological Lattice]]:
:$X$ is [[Definition:Upper Section|upper]] and has [[Definition:Property (S)|property (S)]].
Let $D$ be a [[Definition:Directed Subset|directed subset]] of $S$ such that
:$x \preceq \sup D$
By definition of [[Definition:Upper Section|upper sect... | Infimum of Open Set is Way Below Element in Complete Scott Topological Lattice | https://proofwiki.org/wiki/Infimum_of_Open_Set_is_Way_Below_Element_in_Complete_Scott_Topological_Lattice | https://proofwiki.org/wiki/Infimum_of_Open_Set_is_Way_Below_Element_in_Complete_Scott_Topological_Lattice | [
"Topological Order Theory",
"Way Below Relation"
] | [
"Definition:Complete Lattice",
"Definition:Topological Lattice",
"Definition:Scott Topology",
"Definition:Open Set/Topology",
"Definition:Subset",
"Definition:Element is Way Below"
] | [
"Open iff Upper and with Property (S) in Scott Topological Lattice",
"Definition:Upper Section",
"Definition:Property (S)",
"Definition:Directed Subset",
"Definition:Upper Section",
"Definition:Property (S)",
"Definition:Infimum of Set",
"Definition:Complete Lattice",
"Definition:Lower Bound of Set"... |
proofwiki-12918 | Cyclic Permutations of 5-Digit Multiples of 41 | Let $n$ be a multiple of $41$ with $5$ digits.
Let $m$ be an integer formed by cyclically permuting the digits of $n$.
Then $m$ is a multiple of $41$. | First we note that $10^5 - 1 = 41 \times 271 \times 9$
$10$ generates exactly $5$ elements in $\Z_{41}$ Subgroup Generated by One Element is Cyclic
{{explain|what follows from what? Grammar confusing}}
{{begin-eqn}}
{{eqn | q = \forall k \in \N
| l = 10^{0 + 5 k}
| o = \equiv
| r = 1
| rr= \pmo... | Let $n$ be a [[Definition:Multiple of Integer|multiple]] of $41$ with $5$ [[Definition:Digit|digits]].
Let $m$ be an [[Definition:Integer|integer]] formed by [[Definition:Cyclic Permutation|cyclically permuting]] the [[Definition:Digit|digits]] of $n$.
Then $m$ is a [[Definition:Multiple of Integer|multiple]] of $41... | First we note that $10^5 - 1 = 41 \times 271 \times 9$
$10$ generates exactly $5$ elements in $\Z_{41}$ [[Subgroup Generated by One Element is Cyclic]]
{{explain|what follows from what? Grammar confusing}}
{{begin-eqn}}
{{eqn | q = \forall k \in \N
| l = 10^{0 + 5 k}
| o = \equiv
| r = 1
| r... | Cyclic Permutations of 5-Digit Multiples of 41 | https://proofwiki.org/wiki/Cyclic_Permutations_of_5-Digit_Multiples_of_41 | https://proofwiki.org/wiki/Cyclic_Permutations_of_5-Digit_Multiples_of_41 | [
"Number Theory",
"41"
] | [
"Definition:Multiple/Integer",
"Definition:Digit",
"Definition:Integer",
"Definition:Cyclic Permutation",
"Definition:Digit",
"Definition:Multiple/Integer"
] | [
"Subgroup Generated by One Element is Cyclic",
"Definition:Multiple/Integer",
"Definition:Digit",
"Definition:Integer",
"Definition:Cyclic Permutation",
"Definition:Digit",
"Definition:Multiple/Integer"
] |
proofwiki-12919 | Smallest Prime Number not Difference between Power of 2 and Power of 3 | $41$ is the smallest prime number which is not the difference between a power of $2$ and a power of $3$. | First we have:
{{begin-eqn}}
{{eqn | l = 2
| r = 3^1 - 2^0
| c =
}}
{{eqn | l = 3
| r = 2^2 - 3^0
| c =
}}
{{eqn | l = 5
| r = 3^2 - 2^2
| c =
}}
{{eqn | l = 7
| r = 3^2 - 2^1
| c =
}}
{{eqn | l = 11
| r = 3^3 - 2^4
| c =
}}
{{eqn | l = 13
| r = 2^4... | $41$ is the smallest [[Definition:Prime Number|prime number]] which is not the [[Definition:Integer Subtraction|difference]] between a [[Definition:Integer Power|power]] of $2$ and a [[Definition:Integer Power|power]] of $3$. | First we have:
{{begin-eqn}}
{{eqn | l = 2
| r = 3^1 - 2^0
| c =
}}
{{eqn | l = 3
| r = 2^2 - 3^0
| c =
}}
{{eqn | l = 5
| r = 3^2 - 2^2
| c =
}}
{{eqn | l = 7
| r = 3^2 - 2^1
| c =
}}
{{eqn | l = 11
| r = 3^3 - 2^4
| c =
}}
{{eqn | l = 13
| r = 2^... | Smallest Prime Number not Difference between Power of 2 and Power of 3 | https://proofwiki.org/wiki/Smallest_Prime_Number_not_Difference_between_Power_of_2_and_Power_of_3 | https://proofwiki.org/wiki/Smallest_Prime_Number_not_Difference_between_Power_of_2_and_Power_of_3 | [
"Powers of 2",
"Powers of 3",
"41"
] | [
"Definition:Prime Number",
"Definition:Subtraction/Integers",
"Definition:Power (Algebra)/Integer",
"Definition:Power (Algebra)/Integer"
] | [
"Definition:Integer",
"Congruence of Powers",
"Modulo Multiplication is Well-Defined",
"Definition:Integer",
"Congruence of Powers",
"Definition:Even Integer",
"Congruence of Powers",
"Definition:Even Integer",
"Congruence of Powers",
"Definition:Contradiction",
"Definition:Subtraction/Integers"... |
proofwiki-12920 | Prime-Generating Quadratic of form 2 x squared minus 1000 x minus 2609 | The quadratic function:
:$2 x^2 - 1000 x - 2609$
has $602$ prime values among its first $1000$ values.
Some of those prime numbers are negative. | {{ProofWanted|Lots of tedious calculations needed to prove this.}} | The [[Definition:Quadratic Function|quadratic function]]:
:$2 x^2 - 1000 x - 2609$
has $602$ [[Definition:Prime Number|prime]] [[Definition:Value of Element under Mapping|values]] among its first $1000$ [[Definition:Value of Element under Mapping|values]].
Some of those [[Definition:Prime Number|prime numbers]] are [[... | {{ProofWanted|Lots of tedious calculations needed to prove this.}} | Prime-Generating Quadratic of form 2 x squared minus 1000 x minus 2609 | https://proofwiki.org/wiki/Prime-Generating_Quadratic_of_form_2_x_squared_minus_1000_x_minus_2609 | https://proofwiki.org/wiki/Prime-Generating_Quadratic_of_form_2_x_squared_minus_1000_x_minus_2609 | [
"Polynomial Expressions for Primes"
] | [
"Definition:Quadratic Function",
"Definition:Prime Number",
"Definition:Image (Set Theory)/Mapping/Element",
"Definition:Image (Set Theory)/Mapping/Element",
"Definition:Prime Number",
"Definition:Prime Number/Negative Prime"
] | [] |
proofwiki-12921 | Scott Topology equals to Scott Sigma | Let $\struct {T, \preceq, \tau}$ be a up-complete topological lattice with Scott topology.
Then $\tau = \map \sigma {T, \preceq}$
where $\map \sigma L$ denotes the Scott sigma of $L$. | This follows by Open iff Upper and with Property (S) in Scott Topological Lattice and definition Scott sigma.
{{qed}} | Let $\struct {T, \preceq, \tau}$ be a [[Definition:Up-Complete|up-complete]] [[Definition:Topological Lattice|topological lattice]] with [[Definition:Scott Topology|Scott topology]].
Then $\tau = \map \sigma {T, \preceq}$
where $\map \sigma L$ denotes the [[Definition:Scott Sigma|Scott sigma]] of $L$. | This follows by [[Open iff Upper and with Property (S) in Scott Topological Lattice]] and definition [[Definition:Scott Sigma|Scott sigma]].
{{qed}} | Scott Topology equals to Scott Sigma | https://proofwiki.org/wiki/Scott_Topology_equals_to_Scott_Sigma | https://proofwiki.org/wiki/Scott_Topology_equals_to_Scott_Sigma | [
"Topological Order Theory"
] | [
"Definition:Up-Complete",
"Definition:Topological Lattice",
"Definition:Scott Topology",
"Definition:Scott Sigma"
] | [
"Open iff Upper and with Property (S) in Scott Topological Lattice",
"Definition:Scott Sigma"
] |
proofwiki-12922 | Complement of Lower Closure is Prime Element in Inclusion Ordered Set of Scott Sigma | Let $L = \struct {S, \preceq, \tau}$ be a complete Scott topological lattice.
Let $D = \struct {\map \sigma L, \precsim}$ be an inclusion ordered set of the Scott sigma of $L$.
Let $x \in S$.
Then:
:$\relcomp S {x^\preceq}$ is a prime element in $D$
and:
:$\relcomp S {x^\preceq} \ne S$ | By Scott Topology equals to Scott Sigma:
:$\tau = \map \sigma L$
By Closure of Singleton is Lower Closure of Element in Scott Topological Lattice:
:$x^\preceq = \set x^-$
where $\set x^-$ denotes the topological closure of $\set x$.
By Topological Closure of Singleton is Irreducible:
:$x^\preceq$ is topologically irred... | Let $L = \struct {S, \preceq, \tau}$ be a [[Definition:Complete Lattice|complete]] [[Definition:Scott Topology|Scott]] [[Definition:Topological Lattice|topological lattice]].
Let $D = \struct {\map \sigma L, \precsim}$ be an [[Definition:Inclusion Ordered Set|inclusion ordered set]] of the [[Definition:Scott Sigma|Sco... | By [[Scott Topology equals to Scott Sigma]]:
:$\tau = \map \sigma L$
By [[Closure of Singleton is Lower Closure of Element in Scott Topological Lattice]]:
:$x^\preceq = \set x^-$
where $\set x^-$ denotes the [[Definition:Closure (Topology)|topological closure]] of $\set x$.
By [[Topological Closure of Singleton is Ir... | Complement of Lower Closure is Prime Element in Inclusion Ordered Set of Scott Sigma | https://proofwiki.org/wiki/Complement_of_Lower_Closure_is_Prime_Element_in_Inclusion_Ordered_Set_of_Scott_Sigma | https://proofwiki.org/wiki/Complement_of_Lower_Closure_is_Prime_Element_in_Inclusion_Ordered_Set_of_Scott_Sigma | [
"Prime Elements",
"Topological Order Theory"
] | [
"Definition:Complete Lattice",
"Definition:Scott Topology",
"Definition:Topological Lattice",
"Definition:Inclusion Ordered Set",
"Definition:Scott Sigma",
"Definition:Prime Element (Order Theory)"
] | [
"Scott Topology equals to Scott Sigma",
"Closure of Singleton is Lower Closure of Element in Scott Topological Lattice",
"Definition:Closure (Topology)",
"Topological Closure of Singleton is Irreducible",
"Definition:Irreducible Space",
"Complement of Irreducible Topological Subset is Prime Element",
"D... |
proofwiki-12923 | Topological Closure of Singleton is Irreducible | Let $T = \struct {S, \tau}$ be a topological space.
Let $x$ be a point of $T$.
Then:
:$\set x^-$ is irreducible
where $\set x^-$ denotes the topological closure of $\set x$. | Follows from:
:Trivial Topological Space is Irreducible
:Closure of Irreducible Subspace is Irreducible
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $x$ be a [[Definition:Element|point]] of $T$.
Then:
:$\set x^-$ is [[Definition:Irreducible Space|irreducible]]
where $\set x^-$ denotes the [[Definition:Closure (Topology)|topological closure]] of $\set x$. | Follows from:
:[[Trivial Topological Space is Irreducible]]
:[[Closure of Irreducible Subspace is Irreducible]]
{{qed}} | Topological Closure of Singleton is Irreducible/Proof 1 | https://proofwiki.org/wiki/Topological_Closure_of_Singleton_is_Irreducible | https://proofwiki.org/wiki/Topological_Closure_of_Singleton_is_Irreducible/Proof_1 | [
"Irreducible Spaces",
"Topological Closure of Singleton is Irreducible"
] | [
"Definition:Topological Space",
"Definition:Element",
"Definition:Irreducible Space",
"Definition:Closure (Topology)"
] | [
"Trivial Topological Space is Irreducible",
"Closure of Irreducible Subspace is Irreducible"
] |
proofwiki-12924 | Topological Closure of Singleton is Irreducible | Let $T = \struct {S, \tau}$ be a topological space.
Let $x$ be a point of $T$.
Then:
:$\set x^-$ is irreducible
where $\set x^-$ denotes the topological closure of $\set x$. | {{AimForCont}} that
:$\set x^-$ is not irreducible.
By Set is Subset of its Topological Closure:
:$\set x \subseteq \set x^-$
By definitions of singleton and Subset:
:$x \in \set x^-$
By definition of irreducible:
:$\exists X_1, X_2 \subseteq S: X_1, X_2$ are closed
and:
{{begin-eqn}}
{{eqn | l = \set x^-
| r = X... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $x$ be a [[Definition:Element|point]] of $T$.
Then:
:$\set x^-$ is [[Definition:Irreducible Space|irreducible]]
where $\set x^-$ denotes the [[Definition:Closure (Topology)|topological closure]] of $\set x$. | {{AimForCont}} that
:$\set x^-$ is [[Definition:Not|not]] [[Definition:Irreducible Space|irreducible]].
By [[Set is Subset of its Topological Closure]]:
:$\set x \subseteq \set x^-$
By definitions of [[Definition:Singleton|singleton]] and [[Definition:Subset|Subset]]:
:$x \in \set x^-$
By definition of [[Definition:... | Topological Closure of Singleton is Irreducible/Proof 2 | https://proofwiki.org/wiki/Topological_Closure_of_Singleton_is_Irreducible | https://proofwiki.org/wiki/Topological_Closure_of_Singleton_is_Irreducible/Proof_2 | [
"Irreducible Spaces",
"Topological Closure of Singleton is Irreducible"
] | [
"Definition:Topological Space",
"Definition:Element",
"Definition:Irreducible Space",
"Definition:Closure (Topology)"
] | [
"Definition:Logical Not",
"Definition:Irreducible Space",
"Set is Subset of its Topological Closure",
"Definition:Singleton",
"Definition:Subset",
"Definition:Irreducible Space",
"Definition:Closed Set/Topology",
"Definition:Set Union",
"Definition:Singleton",
"Definition:Subset",
"Definition:Cl... |
proofwiki-12925 | Element equals to Supremum of Infima of Open Sets that Element Belongs implies Topological Lattice is Continuous | Let $L = \left({S, \preceq, \tau}\right)$ be a complete Scott topological lattice.
Let
:$\forall x \in S: x = \sup \left\{ {\inf X: x \in X \in \sigma\left({L}\right)}\right\}$
Then $L$ is continuous. | Thus by Way Below Closure is Directed in Bounded Below Join Semilattice:
:$\forall x \in S:x^\ll$ is directed.
Thus by definition pf complete lattice:
:$L$ is up-complete.
Let $x \in S$.
Define $W := \left\{ {\inf X: x \in X \in \sigma\left({L}\right)}\right\}$
By definition of way below closure:
:$x$ is upper bound fo... | Let $L = \left({S, \preceq, \tau}\right)$ be a [[Definition:Complete Lattice|complete]] [[Definition:Scott Topology|Scott]] [[Definition:Topological Lattice|topological lattice]].
Let
:$\forall x \in S: x = \sup \left\{ {\inf X: x \in X \in \sigma\left({L}\right)}\right\}$
Then $L$ is [[Definition:Continuous Ordered... | Thus by [[Way Below Closure is Directed in Bounded Below Join Semilattice]]:
:$\forall x \in S:x^\ll$ is [[Definition:Directed Subset|directed]].
Thus by definition pf [[Definition:Complete Lattice|complete lattice]]:
:$L$ is [[Definition:Up-Complete|up-complete]].
Let $x \in S$.
Define $W := \left\{ {\inf X: x \in ... | Element equals to Supremum of Infima of Open Sets that Element Belongs implies Topological Lattice is Continuous | https://proofwiki.org/wiki/Element_equals_to_Supremum_of_Infima_of_Open_Sets_that_Element_Belongs_implies_Topological_Lattice_is_Continuous | https://proofwiki.org/wiki/Element_equals_to_Supremum_of_Infima_of_Open_Sets_that_Element_Belongs_implies_Topological_Lattice_is_Continuous | [
"Continuous Lattices",
"Topological Order Theory"
] | [
"Definition:Complete Lattice",
"Definition:Scott Topology",
"Definition:Topological Lattice",
"Definition:Continuous Ordered Set"
] | [
"Way Below Closure is Directed in Bounded Below Join Semilattice",
"Definition:Directed Subset",
"Definition:Complete Lattice",
"Definition:Up-Complete",
"Definition:Way Below Closure",
"Definition:Upper Bound of Set",
"Definition:Supremum of Set",
"Scott Topology equals to Scott Sigma",
"Definition... |
proofwiki-12926 | Proper Class is not Element of Class | Let $\mathrm P$ be a proper class.
Then $\mathrm P$ is not an element of any class.
That is:
:$\neg \exists A : \mathrm P \in A$ | From the definition of a proper class, $\mathrm P$ is not a set.
The result follows by definition of a class.
{{qed}}
Category:Class Theory
6wra5khkj4nktbsc7sdsz5nm70wmsjt | Let $\mathrm P$ be a [[Definition:Proper Class|proper class]].
Then $\mathrm P$ is not an [[Definition:Element of Class|element]] of any [[Definition:Class (Class Theory)|class]].
That is:
:$\neg \exists A : \mathrm P \in A$ | From the definition of a [[Definition:Proper Class|proper class]], $\mathrm P$ is not a [[Definition:Set|set]].
The result follows by definition of a [[Definition:Class (Class Theory)|class]].
{{qed}}
[[Category:Class Theory]]
6wra5khkj4nktbsc7sdsz5nm70wmsjt | Proper Class is not Element of Class | https://proofwiki.org/wiki/Proper_Class_is_not_Element_of_Class | https://proofwiki.org/wiki/Proper_Class_is_not_Element_of_Class | [
"Class Theory"
] | [
"Definition:Class (Class Theory)/Proper Class",
"Definition:Element/Class",
"Definition:Class (Class Theory)"
] | [
"Definition:Class (Class Theory)/Proper Class",
"Definition:Set",
"Definition:Class (Class Theory)",
"Category:Class Theory"
] |
proofwiki-12927 | GCD of Polynomials does not depend on Base Field | Let $E / F$ be a field extension.
Let $P, Q \in F \sqbrk X$ be polynomials.
Let:
:$\gcd \set {P, Q} = R$ in $F \sqbrk X$
:$\gcd \set {P, Q} = S$ in $E \sqbrk X$.
Then $R = S$.
In particular, $S \in F \sqbrk X$. | By definition of greatest common divisor:
:$R \divides S$ in $E \sqbrk X$
By Polynomial Forms over Field is Euclidean Domain, there exist $A, B \in F \sqbrk X$ such that:
:$A P + B Q = R$
Because $S \divides P, Q$:
:$S \divides R$ in $E \sqbrk X$
By $R \divides S$ and $S \divides R$:
:$R = S$
{{qed}}
Category:Field The... | Let $E / F$ be a [[Definition:Field Extension|field extension]].
Let $P, Q \in F \sqbrk X$ be [[Definition:Polynomial (Abstract Algebra)|polynomials]].
Let:
:$\gcd \set {P, Q} = R$ in $F \sqbrk X$
:$\gcd \set {P, Q} = S$ in $E \sqbrk X$.
Then $R = S$.
In particular, $S \in F \sqbrk X$. | By definition of [[Definition:Greatest Common Divisor of Ring Elements|greatest common divisor]]:
:$R \divides S$ in $E \sqbrk X$
By [[Polynomial Forms over Field is Euclidean Domain]], there exist $A, B \in F \sqbrk X$ such that:
:$A P + B Q = R$
Because $S \divides P, Q$:
:$S \divides R$ in $E \sqbrk X$
By $R \div... | GCD of Polynomials does not depend on Base Field | https://proofwiki.org/wiki/GCD_of_Polynomials_does_not_depend_on_Base_Field | https://proofwiki.org/wiki/GCD_of_Polynomials_does_not_depend_on_Base_Field | [
"Field Theory",
"Polynomial Theory",
"Greatest Common Divisor"
] | [
"Definition:Field Extension",
"Definition:Polynomial over Ring"
] | [
"Definition:Greatest Common Divisor/Integral Domain",
"Polynomial Forms over Field is Euclidean Domain",
"Category:Field Theory",
"Category:Polynomial Theory",
"Category:Greatest Common Divisor"
] |
proofwiki-12928 | Prime-Generating Quadratic of form x squared - 79 x + 1601 | The quadratic function:
:$x^2 - 79 x + 1601$
gives prime values for integer $x$ such that $0 \le x \le 79$.
The primes generated are repeated once each. | Let $x = z + 40$.
Then:
{{begin-eqn}}
{{eqn | r = \left({z + 40}\right)^2 - 79 \left({z + 40}\right) + 1601
| o =
| c =
}}
{{eqn | r = z^2 + 2 \times 40 z + 40^2 - 79 z - 79 \times 40 + 1601
| c =
}}
{{eqn | r = z^2 + 80 z + 1600 - 79 z - 3160 + 1601
| c =
}}
{{eqn | r = z^2 + z + 41
|... | The [[Definition:Quadratic Function|quadratic function]]:
:$x^2 - 79 x + 1601$
gives [[Definition:Prime Number|prime]] [[Definition:Value of Element under Mapping|values]] for [[Definition:Integer|integer]] $x$ such that $0 \le x \le 79$.
The [[Definition:Prime Number|primes]] generated are repeated once each. | Let $x = z + 40$.
Then:
{{begin-eqn}}
{{eqn | r = \left({z + 40}\right)^2 - 79 \left({z + 40}\right) + 1601
| o =
| c =
}}
{{eqn | r = z^2 + 2 \times 40 z + 40^2 - 79 z - 79 \times 40 + 1601
| c =
}}
{{eqn | r = z^2 + 80 z + 1600 - 79 z - 3160 + 1601
| c =
}}
{{eqn | r = z^2 + z + 41
... | Prime-Generating Quadratic of form x squared - 79 x + 1601 | https://proofwiki.org/wiki/Prime-Generating_Quadratic_of_form_x_squared_-_79_x_+_1601 | https://proofwiki.org/wiki/Prime-Generating_Quadratic_of_form_x_squared_-_79_x_+_1601 | [
"Polynomial Expressions for Primes",
"Euler Lucky Numbers"
] | [
"Definition:Quadratic Function",
"Definition:Prime Number",
"Definition:Image (Set Theory)/Mapping/Element",
"Definition:Integer",
"Definition:Prime Number"
] | [
"Euler Lucky Number/Examples/41"
] |
proofwiki-12929 | Continuous Group Action is by Homeomorphisms | Let $G$ be a topological group acting continuously on a topological space $X$.
Then $G$ acts by homeomorphisms. | Let $\phi: G \times X \to X$ denote the group action.
Let $g \in G$.
We are given that $\phi$ is a bijection.
The map $\phi_g: X \to X : x \mapsto \map \phi {g, x}$ is continuous because $\phi$ is.
{{explain|Why?}}
By Inverse of Bijection is Bijection, the inverse of $\phi$ is given by $\phi_{g^{-1} }$ is also a biject... | Let $G$ be a [[Definition:Topological Group|topological group]] [[Definition:Continuous Group Action|acting continuously]] on a [[Definition:Topological Space|topological space]] $X$.
Then $G$ [[Definition:Group Action by Homeomorphisms|acts by homeomorphisms]]. | Let $\phi: G \times X \to X$ denote the [[Definition:Group Action|group action]].
Let $g \in G$.
We are [[Definition:Given|given]] that $\phi$ is a [[Definition:Bijection|bijection]].
The map $\phi_g: X \to X : x \mapsto \map \phi {g, x}$ is [[Definition:Continuous Mapping|continuous]] because $\phi$ is.
{{explain|... | Continuous Group Action is by Homeomorphisms | https://proofwiki.org/wiki/Continuous_Group_Action_is_by_Homeomorphisms | https://proofwiki.org/wiki/Continuous_Group_Action_is_by_Homeomorphisms | [
"Topological Group Actions"
] | [
"Definition:Topological Group",
"Definition:Continuous Group Action",
"Definition:Topological Space",
"Definition:Group Action by Homeomorphisms"
] | [
"Definition:Group Action",
"Definition:Given",
"Definition:Bijection",
"Definition:Continuous Mapping",
"Inverse of Bijection is Bijection",
"Definition:Inverse Mapping",
"Definition:Bijection",
"Definition:Inverse Mapping",
"Definition:Continuous Mapping",
"Definition:Inverse Mapping",
"Definit... |
proofwiki-12930 | Discrete Group Acts Continuously iff Acts by Homeomorphisms | Let $G$ be a discrete group acting on a topological space $X$.
Then the following are equivalent:
:$G$ acts continuously
:$G$ acts by homeomorphisms | If $G$ acts continuously, then by Continuous Group Action is by Homeomorphisms, $G$ acts by homeomorphisms
Let $G$ act by homeomorphisms
Let $\phi: G \times X \to X$ denote the group action.
For $g \in G$, denote $\phi_g : X \to X : x \mapsto \map \phi {g, x}$
Let $U \subset X$ be open.
Then:
{{begin-eqn}}
{{eqn | l = ... | Let $G$ be a [[Definition:Discrete Group|discrete group]] [[Definition:Group Action|acting]] on a [[Definition:Topological Space|topological space]] $X$.
Then the following are [[Definition:Logically Equivalent|equivalent]]:
:$G$ [[Definition:Continuous Group Action|acts continuously]]
:$G$ [[Definition:Group Action ... | If $G$ [[Definition:Continuous Group Action|acts continuously]], then by [[Continuous Group Action is by Homeomorphisms]], $G$ [[Definition:Group Action by Homeomorphisms|acts by homeomorphisms]]
Let $G$ [[Definition:Group Action by Homeomorphisms|act by homeomorphisms]]
Let $\phi: G \times X \to X$ denote the [[Def... | Discrete Group Acts Continuously iff Acts by Homeomorphisms | https://proofwiki.org/wiki/Discrete_Group_Acts_Continuously_iff_Acts_by_Homeomorphisms | https://proofwiki.org/wiki/Discrete_Group_Acts_Continuously_iff_Acts_by_Homeomorphisms | [
"Topological Group Actions"
] | [
"Definition:Discrete Group",
"Definition:Group Action",
"Definition:Topological Space",
"Definition:Logical Equivalence",
"Definition:Continuous Group Action",
"Definition:Group Action by Homeomorphisms"
] | [
"Definition:Continuous Group Action",
"Continuous Group Action is by Homeomorphisms",
"Definition:Group Action by Homeomorphisms",
"Definition:Group Action by Homeomorphisms",
"Definition:Group Action",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"... |
proofwiki-12931 | Integers whose Phi times Divisor Count equal Divisor Sum | The positive integers whose Euler $\phi$ function multiplied by its divisor count function equals its divisor sum are:
:$1, 3, 14, 42$
{{OEIS|A104905}} | {{begin-eqn}}
{{eqn | l = \map \phi 1 \map {\sigma_0} 1
| r = 1 \times 1
| c = {{EulerPhiLink|1}}, {{DCFLink|1}}
}}
{{eqn | r = 1
| c =
}}
{{eqn | r = \map {\sigma_1} 1
| c = {{DSFLink|1}}
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map \phi 3 \map {\sigma_0} 3
| r = 2 \times 2
| c = ... | The [[Definition:Positive Integer|positive integers]] whose [[Definition:Euler Phi Function|Euler $\phi$ function]] [[Definition:Integer Multiplication|multiplied by]] its [[Definition:Divisor Count Function|divisor count function]] equals its [[Definition:Divisor Sum Function|divisor sum]] are:
:$1, 3, 14, 42$
{{OEIS... | {{begin-eqn}}
{{eqn | l = \map \phi 1 \map {\sigma_0} 1
| r = 1 \times 1
| c = {{EulerPhiLink|1}}, {{DCFLink|1}}
}}
{{eqn | r = 1
| c =
}}
{{eqn | r = \map {\sigma_1} 1
| c = {{DSFLink|1}}
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map \phi 3 \map {\sigma_0} 3
| r = 2 \times 2
| c ... | Integers whose Phi times Divisor Count equal Divisor Sum | https://proofwiki.org/wiki/Integers_whose_Phi_times_Divisor_Count_equal_Divisor_Sum | https://proofwiki.org/wiki/Integers_whose_Phi_times_Divisor_Count_equal_Divisor_Sum | [
"Euler Phi Function",
"Divisor Count Function",
"Divisor Sum Function"
] | [
"Definition:Positive/Integer",
"Definition:Euler Phi Function",
"Definition:Multiplication/Integers",
"Definition:Divisor Count Function",
"Definition:Divisor Sum Function"
] | [
"Definition:Positive/Integer",
"Definition:Prime Decomposition",
"Euler Phi Function of Integer",
"Divisor Count Function from Prime Decomposition",
"Divisor Sum of Integer",
"Euler Phi Function is Multiplicative",
"Divisor Count Function is Multiplicative",
"Divisor Sum Function is Multiplicative",
... |
proofwiki-12932 | Euler Phi Function of 3 | :$\map \phi 3 = 2$ | From Euler Phi Function of Prime:
:$\map \phi p = p - 1$
As $3$ is a prime number it follows that:
:$\map \phi 3 = 3 - 1 = 2$
{{qed}} | :$\map \phi 3 = 2$ | From [[Euler Phi Function of Prime]]:
:$\map \phi p = p - 1$
As $3$ is a [[Definition:Prime Number|prime number]] it follows that:
:$\map \phi 3 = 3 - 1 = 2$
{{qed}} | Euler Phi Function of 3 | https://proofwiki.org/wiki/Euler_Phi_Function_of_3 | https://proofwiki.org/wiki/Euler_Phi_Function_of_3 | [
"Examples of Euler Phi Function",
"3"
] | [] | [
"Euler Phi Function of Prime",
"Definition:Prime Number"
] |
proofwiki-12933 | Number of Distinct Parenthesizations on Word | Let $w_n$ denote an arbitrary word of $n$ elements.
The number of distinct parenthesizations of $w_n$ is the Catalan number $C_{n - 1}$:
:$C_{n - 1} = \dfrac 1 n \dbinom {2 \paren {n - 1} } {n - 1}$ | Let $w_n$ denote an arbitrary word of $n$ elements.
Let $a_n$ denote the number of ways $W_n$ elements may be parenthesized.
First note that we have:
{{begin-eqn}}
{{eqn | l = a_1
| r = 1
| c =
}}
{{eqn | l = a_2
| r = 1
| c =
}}
{{eqn | l = a_3
| r = 2
| c = that is, $b_1 \paren {... | Let $w_n$ denote an arbitrary [[Definition:Word (Abstract Algebra)|word]] of $n$ [[Definition:Element|elements]].
The number of [[Definition:Distinct|distinct]] [[Definition:Parenthesization|parenthesizations]] of $w_n$ is the [[Definition:Catalan Number|Catalan number]] $C_{n - 1}$:
:$C_{n - 1} = \dfrac 1 n \dbinom {... | Let $w_n$ denote an arbitrary [[Definition:Word (Abstract Algebra)|word]] of $n$ [[Definition:Element|elements]].
Let $a_n$ denote the number of ways $W_n$ [[Definition:Element|elements]] may be [[Definition:Parenthesization|parenthesized]].
First note that we have:
{{begin-eqn}}
{{eqn | l = a_1
| r = 1
... | Number of Distinct Parenthesizations on Word | https://proofwiki.org/wiki/Number_of_Distinct_Parenthesizations_on_Word | https://proofwiki.org/wiki/Number_of_Distinct_Parenthesizations_on_Word | [
"Parenthesization",
"Catalan Numbers"
] | [
"Definition:Word (Abstract Algebra)",
"Definition:Element",
"Definition:Distinct",
"Definition:Parenthesization",
"Definition:Catalan Number"
] | [
"Definition:Word (Abstract Algebra)",
"Definition:Element",
"Definition:Element",
"Definition:Parenthesization",
"Parenthesization/Examples/4",
"Definition:Word (Abstract Algebra)",
"Definition:Element",
"Definition:Sequence",
"Definition:Parenthesization",
"Definition:Parenthesization",
"Defini... |
proofwiki-12934 | Non-Palindromes in Base 2 by Reverse-and-Add Process | Let the number $22$ be expressed in binary: $10110_2$.
When the reverse-and-add process is performed on it repeatedly, it never becomes a palindromic number. | {{begin-eqn}}
{{eqn | l = 10110_2 + 01101_2
| r = 100011_2
| c =
}}
{{eqn | ll= \leadsto
| l = 100011_2 + 110001_2
| r = 1010100_2
| c =
}}
{{end-eqn}}
It remains to be shown that a binary number of this form does not become a palindromic number.
Let $d_n$ denote $n$ repetitions of a bin... | Let the number $22$ be expressed in [[Definition:Binary Notation|binary]]: $10110_2$.
When the [[Definition:Reverse-and-Add|reverse-and-add]] process is performed on it repeatedly, it never becomes a [[Definition:Palindromic Number|palindromic number]]. | {{begin-eqn}}
{{eqn | l = 10110_2 + 01101_2
| r = 100011_2
| c =
}}
{{eqn | ll= \leadsto
| l = 100011_2 + 110001_2
| r = 1010100_2
| c =
}}
{{end-eqn}}
It remains to be shown that a [[Definition:Binary Notation|binary number]] of this form does not become a [[Definition:Palindromic Numb... | Non-Palindromes in Base 2 by Reverse-and-Add Process | https://proofwiki.org/wiki/Non-Palindromes_in_Base_2_by_Reverse-and-Add_Process | https://proofwiki.org/wiki/Non-Palindromes_in_Base_2_by_Reverse-and-Add_Process | [
"Recreational Mathematics",
"Palindromic Numbers",
"Reverse-and-Add"
] | [
"Definition:Binary Notation",
"Definition:Reverse-and-Add",
"Definition:Palindromic Number"
] | [
"Definition:Binary Notation",
"Definition:Palindromic Number",
"Definition:Binary Notation",
"Definition:Number Base",
"Definition:Reverse-and-Add",
"Definition:Palindromic Number"
] |
proofwiki-12935 | Best Approximation from Below to 1 as Sum of Minimal Number of Unit Fractions | :$\dfrac 1 2 + \dfrac 1 3 + \dfrac 1 7 + \dfrac 1 {43} = 1 - \dfrac 1 {1806}$
This is the best approximation from below to $1$ as the minimal sum of unit fractions. | {{ProofWanted|It is not clear exactly what is being stated here.}} | :$\dfrac 1 2 + \dfrac 1 3 + \dfrac 1 7 + \dfrac 1 {43} = 1 - \dfrac 1 {1806}$
This is the best approximation from below to $1$ as the minimal [[Definition:Rational Addition|sum]] of [[Definition:Unit Fraction|unit fractions]]. | {{ProofWanted|It is not clear exactly what is being stated here.}} | Best Approximation from Below to 1 as Sum of Minimal Number of Unit Fractions | https://proofwiki.org/wiki/Best_Approximation_from_Below_to_1_as_Sum_of_Minimal_Number_of_Unit_Fractions | https://proofwiki.org/wiki/Best_Approximation_from_Below_to_1_as_Sum_of_Minimal_Number_of_Unit_Fractions | [
"Unit Fractions"
] | [
"Definition:Addition/Rational Numbers",
"Definition:Unit Fraction"
] | [] |
proofwiki-12936 | Cuboid with Integer Edges and Face Diagonals | The smallest cuboid whose edges and the diagonals of whose faces are all integers has edge lengths $44$, $117$ and $240$.
Its space diagonal, however, is not an integer.
{{expand|Add the definition of Definition:Euler Brick, and rewrite and rename as appropriate.}} | The edges are given as having lengths $44$, $117$ and $240$.
The faces are therefore:
:$44 \times 117$
:$44 \times 240$
:$117 \times 240$
The diagonals of these faces are given by Pythagoras's Theorem as follows:
{{begin-eqn}}
{{eqn | l = 44^2 + 117^2
| r = 15 \, 625
| c =
}}
{{eqn | r = 125^2
| c = ... | The smallest [[Definition:Cuboid|cuboid]] whose [[Definition:Edge of Polyhedron|edges]] and the [[Definition:Diagonal of Parallelogram|diagonals]] of whose [[Definition:Face of Polyhedron|faces]] are all [[Definition:Integer|integers]] has [[Definition:Edge of Polyhedron|edge]] [[Definition:Length of Line|lengths]] $44... | The [[Definition:Edge of Polyhedron|edges]] are given as having [[Definition:Length of Line|lengths]] $44$, $117$ and $240$.
The [[Definition:Face of Polyhedron|faces]] are therefore:
:$44 \times 117$
:$44 \times 240$
:$117 \times 240$
The [[Definition:Diagonal of Parallelogram|diagonals]] of these [[Definition:Face... | Cuboid with Integer Edges and Face Diagonals | https://proofwiki.org/wiki/Cuboid_with_Integer_Edges_and_Face_Diagonals | https://proofwiki.org/wiki/Cuboid_with_Integer_Edges_and_Face_Diagonals | [
"Cuboids"
] | [
"Definition:Cuboid",
"Definition:Polyhedron/Edge",
"Definition:Diameter of Parallelogram",
"Definition:Polyhedron/Face",
"Definition:Integer",
"Definition:Polyhedron/Edge",
"Definition:Linear Measure/Length",
"Definition:Space Diagonal",
"Definition:Integer",
"Definition:Euler Brick"
] | [
"Definition:Polyhedron/Edge",
"Definition:Linear Measure/Length",
"Definition:Polyhedron/Face",
"Definition:Diameter of Parallelogram",
"Definition:Polyhedron/Face",
"Pythagoras's Theorem",
"Definition:Space Diagonal",
"Definition:Integer"
] |
proofwiki-12937 | Largest n such that 1 to n can be Partitioned for no Element to be Sum of 2 Elements in Same Set | $44$ is the largest integer $n$ such that the set of integers from $1$ to $n$ can be partitioned into $4$ subsets such that no integer in any of these subsets is the sum of $2$ other integers in the same subset:
:$\set {1, 3, 5, 15, 17, 19, 26, 28, 40, 42, 44}$
:$\set {2, 7, 8, 18, 21, 24, 27, 33, 37, 38, 43}$
:$\set {... | {{ProofWanted|Outline: find all possible solutions of partitioning $44$ into sum-free sets, then show that $45$ cannot be inserted in any of the sets. The search is, of course, done by computer. Note that $x$ and $2 x$ cannot be in the same set.}} | $44$ is the largest [[Definition:Integer|integer]] $n$ such that the [[Definition:Set|set]] of [[Definition:Integer|integers]] from $1$ to $n$ can be [[Definition:Partition (Set Theory)|partitioned]] into $4$ [[Definition:Subset|subsets]] such that no [[Definition:Integer|integer]] in any of these [[Definition:Subset|s... | {{ProofWanted|Outline: find all possible solutions of partitioning $44$ into sum-free sets, then show that $45$ cannot be inserted in any of the sets. The search is, of course, done by computer. Note that $x$ and $2 x$ cannot be in the same set.}} | Largest n such that 1 to n can be Partitioned for no Element to be Sum of 2 Elements in Same Set | https://proofwiki.org/wiki/Largest_n_such_that_1_to_n_can_be_Partitioned_for_no_Element_to_be_Sum_of_2_Elements_in_Same_Set | https://proofwiki.org/wiki/Largest_n_such_that_1_to_n_can_be_Partitioned_for_no_Element_to_be_Sum_of_2_Elements_in_Same_Set | [
"Recreational Mathematics",
"44"
] | [
"Definition:Integer",
"Definition:Set",
"Definition:Integer",
"Definition:Set Partition",
"Definition:Subset",
"Definition:Integer",
"Definition:Subset",
"Definition:Addition/Integers",
"Definition:Integer",
"Definition:Subset"
] | [] |
proofwiki-12938 | Sequences of 4 Consecutive Integers with Falling Divisor Sum | The following ordered quadruple of consecutive integers have divisor sums which are strictly decreasing:
:$44, 45, 46, 47$
:$104, 105, 106, 107$ | {{begin-eqn}}
{{eqn | l = \map {\sigma_1} {44}
| r = 84
| c = {{DSFLink|44}}
}}
{{eqn | l = \map {\sigma_1} {45}
| r = 78
| c = {{DSFLink|45}}
}}
{{eqn | l = \map {\sigma_1} {46}
| r = 72
| c = {{DSFLink|46}}
}}
{{eqn | l = \map {\sigma_1} {47}
| r = 48
| c = Divisor Sum ... | The following [[Definition:Ordered Quadruple|ordered quadruple]] of consecutive [[Definition:Integer|integers]] have [[Definition:Divisor Sum Function|divisor sums]] which are [[Definition:Strictly Decreasing Mapping|strictly decreasing]]:
:$44, 45, 46, 47$
:$104, 105, 106, 107$ | {{begin-eqn}}
{{eqn | l = \map {\sigma_1} {44}
| r = 84
| c = {{DSFLink|44}}
}}
{{eqn | l = \map {\sigma_1} {45}
| r = 78
| c = {{DSFLink|45}}
}}
{{eqn | l = \map {\sigma_1} {46}
| r = 72
| c = {{DSFLink|46}}
}}
{{eqn | l = \map {\sigma_1} {47}
| r = 48
| c = [[Divisor Su... | Sequences of 4 Consecutive Integers with Falling Divisor Sum | https://proofwiki.org/wiki/Sequences_of_4_Consecutive_Integers_with_Falling_Divisor_Sum | https://proofwiki.org/wiki/Sequences_of_4_Consecutive_Integers_with_Falling_Divisor_Sum | [
"Divisor Sum Function"
] | [
"Definition:Ordered Tuple as Ordered Set/Ordered Quadruple",
"Definition:Integer",
"Definition:Divisor Sum Function",
"Definition:Strictly Decreasing/Mapping"
] | [
"Divisor Sum of Prime Number",
"Definition:Prime Number",
"Divisor Sum of Prime Number",
"Definition:Prime Number"
] |
proofwiki-12939 | Pairs of Consecutive Integers with 6 Divisors | The following sequence of integers are those $n$ which fulfil the equation:
:$\map {\sigma_0} n = \map {\sigma_0} {n + 1} = 6$
where $\map {\sigma_0} n$ denotes the divisor count function.
That is, they are the first of pairs of consecutive integers which each have $6$ divisors:
:$44, 75, 98, 116, 147, 171, 242, 243, ... | From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
{{begin-eqn}}
{{eqn |... | The following [[Definition:Integer Sequence|sequence]] of [[Definition:Positive Integer|integers]] are those $n$ which fulfil the equation:
:$\map {\sigma_0} n = \map {\sigma_0} {n + 1} = 6$
where $\map {\sigma_0} n$ denotes the [[Definition:Divisor Count Function|divisor count function]].
That is, they are the first... | From [[Divisor Count Function from Prime Decomposition]]:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of [[Definition:Distinct|distinct]] [[Definition:Prime Factor|prime factors]] in the [[Definition:Prime Decomposition|prime decomposition]] of $n$
:$k_j$ denotes t... | Pairs of Consecutive Integers with 6 Divisors | https://proofwiki.org/wiki/Pairs_of_Consecutive_Integers_with_6_Divisors | https://proofwiki.org/wiki/Pairs_of_Consecutive_Integers_with_6_Divisors | [
"Divisor Count Function"
] | [
"Definition:Integer Sequence",
"Definition:Positive/Integer",
"Definition:Divisor Count Function",
"Definition:Ordered Pair",
"Definition:Positive/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Divisor Count Function from Prime Decomposition",
"Definition:Distinct",
"Definition:Prime Factor",
"Definition:Prime Decomposition",
"Definition:Prime Decomposition/Multiplicity",
"Definition:Prime Number",
"Definition:Prime Decomposition"
] |
proofwiki-12940 | Hermitian Conjugate is Involution | Let $\mathbf A$ be a complex-valued matrix.
Let $\mathbf A^*$ denote the Hermitian conjugate of $\mathbf A$.
Then the operation of Hermitian conjugate is an involution:
:$\paren {\mathbf A^*}^* = \mathbf A$ | {{begin-eqn}}
{{eqn | l = \sqbrk {\paren {\mathbf A^*}^* }_{i j}
| r = \overline {\sqbrk {\mathbf A^*}_{j i} }
| c = {{Defof|Hermitian Conjugate}}
}}
{{eqn | r = \overline {\paren {\overline {\sqbrk {\mathbf A}_{i j} } } }
| c = {{Defof|Hermitian Conjugate}}
}}
{{eqn | r = \sqbrk {\mathbf A}_{i j}
... | Let $\mathbf A$ be a [[Definition:Complex Number|complex-valued]] [[Definition:Matrix|matrix]].
Let $\mathbf A^*$ denote the [[Definition:Hermitian Conjugate|Hermitian conjugate]] of $\mathbf A$.
Then the [[Definition:Unary Operation|operation]] of [[Definition:Hermitian Conjugate|Hermitian conjugate]] is an [[Defin... | {{begin-eqn}}
{{eqn | l = \sqbrk {\paren {\mathbf A^*}^* }_{i j}
| r = \overline {\sqbrk {\mathbf A^*}_{j i} }
| c = {{Defof|Hermitian Conjugate}}
}}
{{eqn | r = \overline {\paren {\overline {\sqbrk {\mathbf A}_{i j} } } }
| c = {{Defof|Hermitian Conjugate}}
}}
{{eqn | r = \sqbrk {\mathbf A}_{i j}
... | Hermitian Conjugate is Involution | https://proofwiki.org/wiki/Hermitian_Conjugate_is_Involution | https://proofwiki.org/wiki/Hermitian_Conjugate_is_Involution | [
"Hermitian Conjugates",
"Involutions"
] | [
"Definition:Complex Number",
"Definition:Matrix",
"Definition:Hermitian Conjugate",
"Definition:Operation/Unary Operation",
"Definition:Hermitian Conjugate",
"Definition:Involution (Mapping)"
] | [
"Complex Conjugation is Involution",
"Category:Hermitian Conjugates",
"Category:Involutions"
] |
proofwiki-12941 | Number as Sum of Distinct Primes greater than 11 | Every number greater than $45$ can be expressed as the sum of distinct primes greater than $11$. | Let $S = \set {s_n}_{n \mathop \in N}$ be the set of primes greater than $11$ ordered by size.
Then $S = \set {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, \dots}$.
By Bertrand-Chebyshev Theorem:
:$s_{n + 1} \le 2 s_n$ for all $n \in \N$.
We observe that every integer $n$ where $45 < n \le 45 + s_{11} = 92$ can be expressed... | Every number greater than $45$ can be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct|distinct]] [[Definition:Prime Number|primes]] greater than $11$. | Let $S = \set {s_n}_{n \mathop \in N}$ be the set of [[Definition:Prime Number|primes]] greater than $11$ ordered by size.
Then $S = \set {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, \dots}$.
By [[Bertrand-Chebyshev Theorem]]:
:$s_{n + 1} \le 2 s_n$ for all $n \in \N$.
We observe that every [[Definition:Integer|integer... | Number as Sum of Distinct Primes greater than 11 | https://proofwiki.org/wiki/Number_as_Sum_of_Distinct_Primes_greater_than_11 | https://proofwiki.org/wiki/Number_as_Sum_of_Distinct_Primes_greater_than_11 | [
"Prime Numbers"
] | [
"Definition:Addition/Integers",
"Definition:Distinct",
"Definition:Prime Number"
] | [
"Definition:Prime Number",
"Bertrand-Chebyshev Theorem",
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Distinct",
"Richert's Theorem"
] |
proofwiki-12942 | Subclass of Set is Set | Let $A$ be a set.
Let $\map \phi x$ be a condition in which $x$ is taken to be a set.
Then there exists a set that consists of all of the elements of $A$ that satisfies this condition.
In ZF, this result is known as the Axiom of Specification. | By the axiom of class comprehension, let $B$ be the class defined as:
{{begin-eqn}}
{{eqn | l = B
| r = \set {x: x \in A \land \map \phi x}
}}
{{eqn | r = \set {x \in A: \map \phi x}
| c = Set-Builder Notation
}}
{{end-eqn}}
{{AimForCont}} that $B$ is not a set.
Then $B$ must be a proper class.
It is easily... | Let $A$ be a [[Definition:Set|set]].
Let $\map \phi x$ be a [[Definition:Propositional Function|condition]] in which $x$ is taken to be a [[Definition:Set|set]].
Then there exists a [[Definition:Set|set]] that consists of all of the [[Definition:Element|elements]] of $A$ that satisfies this [[Definition:Propositiona... | By the [[Axiom:Class Comprehension Schema|axiom of class comprehension]], let $B$ be the [[Definition:Class (Class Theory)|class]] defined as:
{{begin-eqn}}
{{eqn | l = B
| r = \set {x: x \in A \land \map \phi x}
}}
{{eqn | r = \set {x \in A: \map \phi x}
| c = [[Definition:Set-Builder Notation|Set-Builder... | Subclass of Set is Set | https://proofwiki.org/wiki/Subclass_of_Set_is_Set | https://proofwiki.org/wiki/Subclass_of_Set_is_Set | [
"Gödel-Bernays Class Theory"
] | [
"Definition:Set",
"Definition:Propositional Function",
"Definition:Set",
"Definition:Set",
"Definition:Element",
"Definition:Propositional Function",
"Definition:Zermelo-Fraenkel Set Theory",
"Axiom:Axiom of Specification/Class Theory"
] | [
"Axiom:Class Comprehension Schema",
"Definition:Class (Class Theory)",
"Definition:Set/Definition by Predicate",
"Definition:Set",
"Definition:Class (Class Theory)/Proper Class",
"Axiom:Axiom of Powers/Class Theory",
"Definition:Power Set",
"Proper Class is not Element of Class",
"Definition:Contrad... |
proofwiki-12943 | Hexagonal Number as 4 times Triangular Number plus n | Let $H_n$ be the $n$th hexagonal number.
Then:
:$H_n = 4 T_{n - 1} + n$
where $T_{n - 1}$ is the $n - 1$th triangular number. | {{begin-eqn}}
{{eqn | l = H_n
| r = n \paren {2 n - 1}
| c = Closed Form for Hexagonal Numbers
}}
{{eqn | r = \frac {2 n \paren {2 n - 1} } 2
| c =
}}
{{eqn | r = \frac {2 n \paren {2 n - 2 + 1} } 2
| c =
}}
{{eqn | r = 4 \frac {n \paren {n - 1} } 2 + \frac {2 n} 2
| c =
}}
{{eqn | r = ... | Let $H_n$ be the $n$th [[Definition:Hexagonal Number|hexagonal number]].
Then:
:$H_n = 4 T_{n - 1} + n$
where $T_{n - 1}$ is the $n - 1$th [[Definition:Triangular Number|triangular number]]. | {{begin-eqn}}
{{eqn | l = H_n
| r = n \paren {2 n - 1}
| c = [[Closed Form for Hexagonal Numbers]]
}}
{{eqn | r = \frac {2 n \paren {2 n - 1} } 2
| c =
}}
{{eqn | r = \frac {2 n \paren {2 n - 2 + 1} } 2
| c =
}}
{{eqn | r = 4 \frac {n \paren {n - 1} } 2 + \frac {2 n} 2
| c =
}}
{{eqn | ... | Hexagonal Number as 4 times Triangular Number plus n | https://proofwiki.org/wiki/Hexagonal_Number_as_4_times_Triangular_Number_plus_n | https://proofwiki.org/wiki/Hexagonal_Number_as_4_times_Triangular_Number_plus_n | [
"Hexagonal Numbers",
"Triangular Numbers"
] | [
"Definition:Hexagonal Number",
"Definition:Triangular Number"
] | [
"Closed Form for Hexagonal Numbers",
"Closed Form for Triangular Numbers"
] |
proofwiki-12944 | Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite/Dependent on N Functions | Let $K$ be a (real) functional, such that:
:$\ds K \sqbrk {\mathbf h} = \int_a^b \paren {\mathbf h' \mathbf P \mathbf h' + \mathbf h \mathbf Q \mathbf h} \rd x$
where:
:$\mathbf h$ is an $N$-dimensional vector
:$\mathbf Q$ is a $N \times N$ matrix
:$\mathbf P$ is a $N\times N$ symmetric positive definite matrix.
Let $\... | === Necessary Condition ===
Let $\mathbf W$ be an arbitrary differentiable symmetric matrix.
Then
{{begin-eqn}}
{{eqn | l = 0
| r = \int_a^b \map {\frac \d {\d x} } {\mathbf h \mathbf W \mathbf h} \rd x
}}
{{eqn | r = \int_a^b \mathbf h \mathbf W' \mathbf h \rd x + 2 \int_a^b \mathbf h' \mathbf W \mathbf h \rd x
... | Let $K$ be a [[Definition:Real Functional|(real) functional]], such that:
:$\ds K \sqbrk {\mathbf h} = \int_a^b \paren {\mathbf h' \mathbf P \mathbf h' + \mathbf h \mathbf Q \mathbf h} \rd x$
where:
:$\mathbf h$ is an [[Definition:Dimension|$N$-dimensional]] [[Definition:Vector|vector]]
:$\mathbf Q$ is a $N \times N$... | === Necessary Condition ===
Let $\mathbf W$ be an arbitrary [[Definition:Differentiable Symmetric Matrix|differentiable]] [[Definition:Symmetric Matrix|symmetric matrix]].
Then
{{begin-eqn}}
{{eqn | l = 0
| r = \int_a^b \map {\frac \d {\d x} } {\mathbf h \mathbf W \mathbf h} \rd x
}}
{{eqn | r = \int_a^b \math... | Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite/Dependent on N Functions | https://proofwiki.org/wiki/Necessary_and_Sufficient_Condition_for_Quadratic_Functional_to_be_Positive_Definite/Dependent_on_N_Functions | https://proofwiki.org/wiki/Necessary_and_Sufficient_Condition_for_Quadratic_Functional_to_be_Positive_Definite/Dependent_on_N_Functions | [
"Calculus of Variations"
] | [
"Definition:Functional/Real",
"Definition:Dimension",
"Definition:Vector",
"Definition:Symmetric Matrix",
"Definition:Positive Definite",
"Definition:Matrix",
"Definition:Point",
"Definition:Conjugate Point (Calculus of Variations)"
] | [
"Definition:Differentiable Symmetric Matrix",
"Definition:Symmetric Matrix",
"Definition:Conjugate Point (Calculus of Variations)",
"Definition:Conjugate Point (Calculus of Variations)",
"lemma",
"Definition:Conjugate Point (Calculus of Variations)"
] |
proofwiki-12945 | Set of Upper Closures of Compact Elements is Basis implies Complete Scott Topological Lattice is Algebraic | Let $L = \struct {S, \preceq, \tau}$ be a complete Scott Definition:Topological Lattice.
Let $\BB = \set {x^\succeq: x \in \map K L}$ be a basis of $L$, where:
:$x^\succeq$ denotes the upper closure of $x$,
:$\map K L$ denotes the compact subset of $L$.
Then $L$ is algebraic. | Thus by Compact Closure is Directed:
:$\forall x \in S:x^{\mathrm {compact} }$ is directed
where $x^{\mathrm {compact} }$ denotes the compact closure of $x$.
Thus by definition of complete lattice:
:$L$ is up-complete.
Let $x \in S$.
By definition of lower closure of element:
:$x$ is upper closure for $x^\preceq$
By de... | Let $L = \struct {S, \preceq, \tau}$ be a [[Definition:Complete Lattice|complete]] [[Definition:Scott Topology|Scott]] [[Definition:Topological Lattice]].
Let $\BB = \set {x^\succeq: x \in \map K L}$ be a [[Definition:Analytic Basis|basis]] of $L$, where:
:$x^\succeq$ denotes the [[Definition:Upper Closure of Element|... | Thus by [[Compact Closure is Directed]]:
:$\forall x \in S:x^{\mathrm {compact} }$ is [[Definition:Directed Subset|directed]]
where $x^{\mathrm {compact} }$ denotes the [[Definition:Compact Closure|compact closure]] of $x$.
Thus by definition of [[Definition:Complete Lattice|complete lattice]]:
:$L$ is [[Definition:Up... | Set of Upper Closures of Compact Elements is Basis implies Complete Scott Topological Lattice is Algebraic | https://proofwiki.org/wiki/Set_of_Upper_Closures_of_Compact_Elements_is_Basis_implies_Complete_Scott_Topological_Lattice_is_Algebraic | https://proofwiki.org/wiki/Set_of_Upper_Closures_of_Compact_Elements_is_Basis_implies_Complete_Scott_Topological_Lattice_is_Algebraic | [
"Continuous Lattices",
"Topological Order Theory"
] | [
"Definition:Complete Lattice",
"Definition:Scott Topology",
"Definition:Topological Lattice",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Upper Closure/Element",
"Definition:Compact Subset of Lattice",
"Definition:Algebraic Ordered Set"
] | [
"Compact Closure is Directed",
"Definition:Directed Subset",
"Definition:Compact Closure",
"Definition:Complete Lattice",
"Definition:Up-Complete",
"Definition:Lower Closure/Element",
"Definition:Upper Bound of Set",
"Definition:Supremum of Set",
"Compact Closure is Intersection of Lower Closure and... |
proofwiki-12946 | Class is Proper iff Bijection from Class to Proper Class | Let $A$ be a class.
Let $\mathrm P$ be a proper class.
Then $A$ is proper {{iff}} there exists a bijection from $A$ to $\mathrm P$. | {{NotZFC}} | Let $A$ be a [[Definition:Class (Class Theory)|class]].
Let $\mathrm P$ be a [[Definition:Proper Class|proper class]].
Then $A$ is [[Definition:Proper Class|proper]] {{iff}} there exists a [[Definition:Class Bijection|bijection]] from $A$ to $\mathrm P$. | {{NotZFC}} | Class is Proper iff Bijection from Class to Proper Class | https://proofwiki.org/wiki/Class_is_Proper_iff_Bijection_from_Class_to_Proper_Class | https://proofwiki.org/wiki/Class_is_Proper_iff_Bijection_from_Class_to_Proper_Class | [
"Gödel-Bernays Class Theory",
"Class Mappings"
] | [
"Definition:Class (Class Theory)",
"Definition:Class (Class Theory)/Proper Class",
"Definition:Class (Class Theory)/Proper Class",
"Definition:Bijection/Class Theory"
] | [] |
proofwiki-12947 | Numbers which Multiplied by 2 are the Reverse of when Added to 2 | {{begin-eqn}}
{{eqn | l = 47 + 2
| r = 49
}}
{{eqn | l = 47 \times 2
| r = 94
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 497 + 2
| r = 499
}}
{{eqn | l = 497 \times 2
| r = 994
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 4997 + 2
| r = 4999
}}
{{eqn | l = 4997 \times 2
| r = 9994
}}
{{end... | We have that:
:$\ds \paren {4 \times 10^n + \sum_{k \mathop = 1}^{n - 1} 9 \times 10^k + 7} + 2 = 4 \times 10^n + \sum_{k \mathop = 0}^{n - 1} 9 \times 10^k$
using the Basis Representation Theorem.
It remains to be demonstrated that:
:$\ds 2 \times \paren {4 \times 10^n + \sum_{k \mathop = 1}^{n - 1} 9 \times 10^k + 7}... | {{begin-eqn}}
{{eqn | l = 47 + 2
| r = 49
}}
{{eqn | l = 47 \times 2
| r = 94
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 497 + 2
| r = 499
}}
{{eqn | l = 497 \times 2
| r = 994
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 4997 + 2
| r = 4999
}}
{{eqn | l = 4997 \times 2
| r = 9994
}}
{... | We have that:
:$\ds \paren {4 \times 10^n + \sum_{k \mathop = 1}^{n - 1} 9 \times 10^k + 7} + 2 = 4 \times 10^n + \sum_{k \mathop = 0}^{n - 1} 9 \times 10^k$
using the [[Basis Representation Theorem]].
It remains to be demonstrated that:
:$\ds 2 \times \paren {4 \times 10^n + \sum_{k \mathop = 1}^{n - 1} 9 \times 10... | Numbers which Multiplied by 2 are the Reverse of when Added to 2 | https://proofwiki.org/wiki/Numbers_which_Multiplied_by_2_are_the_Reverse_of_when_Added_to_2 | https://proofwiki.org/wiki/Numbers_which_Multiplied_by_2_are_the_Reverse_of_when_Added_to_2 | [
"Recreational Mathematics",
"Reversals"
] | [] | [
"Basis Representation Theorem",
"Basis Representation Theorem"
] |
proofwiki-12948 | Infinite Product of Analytic Functions | Let $D \subset \C$ be an open connected set.
Let $\sequence {f_n}$ be a sequence of analytic functions $f_n: D \to \C$ that are not identically zero.
Let $\ds \sum_{n \mathop = 1}^\infty \paren {f_n - 1}$ converge locally uniformly absolutely on $D$.
Then:
:$(1): \quad f = \ds \prod_{n \mathop = 1}^\infty f_n$ converge... | Because $\ds \sum_{n \mathop = 1}^\infty \paren {f_n - 1}$ converges locally uniformly absolutely on $D$, the series converges locally uniformly absolutely.
Thus $f = \ds \prod_{n \mathop = 1}^\infty f_n$ converges locally uniformly absolutely.
By Uniform Limit of Analytic Functions is Analytic, $f$ is analytic
The las... | Let $D \subset \C$ be an [[Definition:Open Set (Complex Analysis)|open]] [[Definition:Connected Topological Space|connected set]].
Let $\sequence {f_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Analytic Function|analytic functions]] $f_n: D \to \C$ that are not [[Definition:Identically Zero|identically ze... | Because $\ds \sum_{n \mathop = 1}^\infty \paren {f_n - 1}$ [[Definition:Locally Uniform Absolute Convergence|converges locally uniformly absolutely]] on $D$, the series [[Definition:Local Uniform Absolute Convergence|converges locally uniformly absolutely]].
Thus $f = \ds \prod_{n \mathop = 1}^\infty f_n$ [[Definition... | Infinite Product of Analytic Functions | https://proofwiki.org/wiki/Infinite_Product_of_Analytic_Functions | https://proofwiki.org/wiki/Infinite_Product_of_Analytic_Functions | [
"Infinite Products"
] | [
"Definition:Open Set/Complex Analysis",
"Definition:Connected Topological Space",
"Definition:Sequence",
"Definition:Analytic Function",
"Definition:Identically Zero",
"Definition:Locally Uniform Absolute Convergence",
"Definition:Locally Uniform Absolute Convergence of Product",
"Definition:Analytic ... | [
"Definition:Locally Uniform Absolute Convergence",
"Definition:Locally Uniform Absolute Convergence",
"Definition:Locally Uniform Absolute Convergence of Product",
"Uniform Limit of Analytic Functions is Analytic",
"Definition:Analytic Function",
"Zeroes of Infinite Product of Analytic Functions"
] |
proofwiki-12949 | Prime between n and 9 n divided by 8 | Let $n \in \Z$ be an integer such that $n \ge 48$.
Then there exists a prime number $p$ such that $n < p < \dfrac {9 n} 8$. | Let $\map P n$ be the property:
:there exists a prime number $p$ such that $n \le p \le \dfrac {9 n} 8$.
First note that $\map P n$ does not hold for the following $n < 48$:
:$0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 15, 19, 21, 23, 24, 25, 31, 32, 47$
Taking $47$ as an example:
{{begin-eqn}}
{{eqn | l = 48
| r ... | Let $n \in \Z$ be an [[Definition:Integer|integer]] such that $n \ge 48$.
Then there exists a [[Definition:Prime Number|prime number]] $p$ such that $n < p < \dfrac {9 n} 8$. | Let $\map P n$ be the [[Definition:Propositional Function|property]]:
:there exists a [[Definition:Prime Number|prime number]] $p$ such that $n \le p \le \dfrac {9 n} 8$.
First note that $\map P n$ does not hold for the following $n < 48$:
:$0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 15, 19, 21, 23, 24, 25, 31, 32, 47... | Prime between n and 9 n divided by 8 | https://proofwiki.org/wiki/Prime_between_n_and_9_n_divided_by_8 | https://proofwiki.org/wiki/Prime_between_n_and_9_n_divided_by_8 | [
"Prime Numbers"
] | [
"Definition:Integer",
"Definition:Prime Number"
] | [
"Definition:Propositional Function",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number"
] |
proofwiki-12950 | Uniformly Continuous Function Preserves Uniform Convergence | Let $X$ be a set.
Let $M = \struct {A_1, d_1}$ and $N = \struct {A_2, d_2}$ be metric spaces.
Let $\sequence {g_n}$ be a sequence of mappings $g_n: X \to A_1$.
Let $g_n$ converge to $g$ uniformly on $X$ as $n \to \infty$.
Let $f: A_1 \to A_2$ be uniformly continuous on $A_1$.
Then the sequence $\sequence {f \circ g_n}... | Let $\epsilon \in \R_{>0}$.
Because $f$ is uniformly continuous on $A_1$, there exists $\delta \in \R_{>0}$ such that:
:$\forall x, y \in A_1: \map {d_2} {\map f x, \map f y} < \epsilon$ for $\map {d_1} {x, y} < \delta$
Because $\sequence {g_n}$ converges to $g$ uniformly on $X$ as $n \to \infty$, there exists $N > 0$ ... | Let $X$ be a [[Definition:Set|set]].
Let $M = \struct {A_1, d_1}$ and $N = \struct {A_2, d_2}$ be [[Definition:Metric Space|metric spaces]].
Let $\sequence {g_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Mapping|mappings]] $g_n: X \to A_1$.
Let $g_n$ [[Definition:Uniform Convergence|converge to $g$ unif... | Let $\epsilon \in \R_{>0}$.
Because $f$ is [[Definition:Uniformly Continuous Mapping (Metric Spaces)|uniformly continuous]] on $A_1$, there exists $\delta \in \R_{>0}$ such that:
:$\forall x, y \in A_1: \map {d_2} {\map f x, \map f y} < \epsilon$ for $\map {d_1} {x, y} < \delta$
Because $\sequence {g_n}$ [[Definition... | Uniformly Continuous Function Preserves Uniform Convergence | https://proofwiki.org/wiki/Uniformly_Continuous_Function_Preserves_Uniform_Convergence | https://proofwiki.org/wiki/Uniformly_Continuous_Function_Preserves_Uniform_Convergence | [
"Uniform Convergence",
"Uniform Continuity"
] | [
"Definition:Set",
"Definition:Metric Space",
"Definition:Sequence",
"Definition:Mapping",
"Definition:Uniform Convergence",
"Definition:Uniform Continuity/Metric Space",
"Definition:Sequence",
"Definition:Uniform Convergence"
] | [
"Definition:Uniform Continuity/Metric Space",
"Definition:Uniform Convergence",
"Definition:Sequence",
"Definition:Uniform Convergence"
] |
proofwiki-12951 | Complex Exponential is Uniformly Continuous on Half-Planes | Let $a \in \R$.
Then $\exp$ is uniformly continuous on the half-plane $\set {z \in \C : \map \Re z \le a}$. | Let $\epsilon > 0$.
For $x, y \in \C$ with $\map \Re x, \map \Re y \le a$:
{{begin-eqn}}
{{eqn | l = \cmod {e^x - e^y}
| r = \cmod {e^y} \cdot \cmod {e^{x - y} - 1}
}}
{{eqn | r = e^{\map \Re y} \cdot \cmod {e^{x - y} - 1}
| c = Modulus of Exponential is Exponential of Real Part
}}
{{eqn | o = \le
| r... | Let $a \in \R$.
Then $\exp$ is [[Definition:Uniform Continuity|uniformly continuous]] on the [[Definition:Half-Plane|half-plane]] $\set {z \in \C : \map \Re z \le a}$. | Let $\epsilon > 0$.
For $x, y \in \C$ with $\map \Re x, \map \Re y \le a$:
{{begin-eqn}}
{{eqn | l = \cmod {e^x - e^y}
| r = \cmod {e^y} \cdot \cmod {e^{x - y} - 1}
}}
{{eqn | r = e^{\map \Re y} \cdot \cmod {e^{x - y} - 1}
| c = [[Modulus of Exponential is Exponential of Real Part]]
}}
{{eqn | o = \le
... | Complex Exponential is Uniformly Continuous on Half-Planes | https://proofwiki.org/wiki/Complex_Exponential_is_Uniformly_Continuous_on_Half-Planes | https://proofwiki.org/wiki/Complex_Exponential_is_Uniformly_Continuous_on_Half-Planes | [
"Exponential Function",
"Uniform Continuity"
] | [
"Definition:Uniform Continuity",
"Definition:Half-Plane"
] | [
"Modulus of Exponential is Exponential of Real Part",
"Exponential is Strictly Increasing",
"Exponential Function is Continuous",
"Definition:Uniform Continuity",
"Category:Exponential Function",
"Category:Uniform Continuity"
] |
proofwiki-12952 | Local Uniform Convergence Implies Compact Convergence | Let $X$ be a topological space.
Let $M = \struct {A, d}$ be a metric space.
Let $\sequence {f_n}$ be a sequence of mappings $f_n : X \to M$.
Let $f_n$ converge locally uniformly to $f : X \to M$.
Then $f_n$ converges compactly to $f$. | {{Proofread}}
{{tidy}}
{{MissingLinks}}
Recall that a sequence $f_n$ converges compactly to $f$, if $f_n$ converges uniformly to $f$ over any compact subset $K \subset X$.
Consider a sequence $f_n$ which converges locally uniformly to $f : X \to M$.f
Let $K$ be an arbitrary subset $K \subset X$.
For $x \in K$ there exi... | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $\sequence {f_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Mapping|mappings]] $f_n : X \to M$.
Let $f_n$ [[Definition:Locally Uniform Convergence|converge locally ... | {{Proofread}}
{{tidy}}
{{MissingLinks}}
Recall that a sequence $f_n$ [[Definition:Compact Convergence|converges compactly]] to $f$, if $f_n$ [[Definition:Uniform Convergence|converges uniformly]] to $f$ over any [[Definition:Compact Subspace|compact subset]] $K \subset X$.
Consider a sequence $f_n$ which [[Definition... | Local Uniform Convergence Implies Compact Convergence | https://proofwiki.org/wiki/Local_Uniform_Convergence_Implies_Compact_Convergence | https://proofwiki.org/wiki/Local_Uniform_Convergence_Implies_Compact_Convergence | [
"Uniform Convergence",
"Compact Convergence"
] | [
"Definition:Topological Space",
"Definition:Metric Space",
"Definition:Sequence",
"Definition:Mapping",
"Definition:Locally Uniform Convergence",
"Definition:Compact Convergence"
] | [
"Definition:Compact Convergence",
"Definition:Uniform Convergence",
"Definition:Compact Topological Space/Subspace",
"Definition:Locally Uniform Convergence",
"Definition:Open Neighborhood",
"Definition:Uniform Convergence",
"Definition:Open Cover/Subset",
"Definition:Compact Topological Space",
"De... |
proofwiki-12953 | Compact Convergence Implies Local Uniform Convergence if Weakly Locally Compact | Let $T = \struct {S, \tau}$ be a weakly locally compact topological space.
Let $M = \struct {A, d}$ be a metric space.
Let $\sequence {f_n}$ be a sequence of mappings $f_n: X \to M$.
Let $f_n$ converge compactly to $f: X \to M$.
Then $f_n$ converges locally uniformly to $f$. | Because $T$ is weakly locally compact, every point of $S$ has a compact neighborhood in $T$.
We are given $f_n$ converge compactly to $f$, so $f_n$ converges uniformly to $f$ on every compact subset of $X$
So every point of $S$ has a (compact) neighborhood on which $f_n$ converges uniformly to $f$.
Thus $f_n$ converges... | Let $T = \struct {S, \tau}$ be a [[Definition:Weakly Locally Compact Space|weakly locally compact topological space]].
Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $\sequence {f_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Mapping|mappings]] $f_n: X \to M$.
Let $f_n$ [[Def... | Because $T$ is [[Definition:Weakly Locally Compact Space|weakly locally compact]], every point of $S$ has a [[Definition:Compact Topological Subspace|compact]] [[Definition:Neighborhood of Point|neighborhood]] in $T$.
We are [[Definition:Given|given]] $f_n$ [[Definition:Compact Convergence|converge compactly]] to $f$,... | Compact Convergence Implies Local Uniform Convergence if Weakly Locally Compact | https://proofwiki.org/wiki/Compact_Convergence_Implies_Local_Uniform_Convergence_if_Weakly_Locally_Compact | https://proofwiki.org/wiki/Compact_Convergence_Implies_Local_Uniform_Convergence_if_Weakly_Locally_Compact | [
"Uniform Convergence",
"Compact Convergence"
] | [
"Definition:Weakly Locally Compact Space",
"Definition:Metric Space",
"Definition:Sequence",
"Definition:Mapping",
"Definition:Compact Convergence",
"Definition:Locally Uniform Convergence"
] | [
"Definition:Weakly Locally Compact Space",
"Definition:Compact Topological Space/Subspace",
"Definition:Neighborhood (Topology)/Point",
"Definition:Given",
"Definition:Compact Convergence",
"Definition:Uniform Convergence",
"Definition:Compact Topological Space/Subspace",
"Definition:Compact Topologic... |
proofwiki-12954 | Smallest Pair of Quasiamicable Numbers | The smallest pair of quasiamicable numbers is $48$ and $75$. | From Quasiamicable Numbers: $48$ and $75$ we have that $48$ and $75$ are quasiamicable numbers.
It remains to be demonstrated that these are indeed the smallest such.
Let $n$ be the smaller number of the quasiamicable pair.
Then we must have $\map {\sigma_1} n - n - 1 > n$.
Since $\map {\sigma_1} n > 2 n$, $n$ is abund... | The smallest [[Definition:Doubleton|pair]] of [[Definition:Quasiamicable Numbers|quasiamicable numbers]] is $48$ and $75$. | From [[Quasiamicable Numbers/Examples/48,75|Quasiamicable Numbers: $48$ and $75$]] we have that $48$ and $75$ are [[Definition:Quasiamicable Numbers|quasiamicable numbers]].
It remains to be demonstrated that these are indeed the smallest such.
Let $n$ be the smaller number of the [[Definition:Quasiamicable Numbers|... | Smallest Pair of Quasiamicable Numbers | https://proofwiki.org/wiki/Smallest_Pair_of_Quasiamicable_Numbers | https://proofwiki.org/wiki/Smallest_Pair_of_Quasiamicable_Numbers | [
"Quasiamicable Numbers",
"48",
"75"
] | [
"Definition:Doubleton",
"Definition:Quasiamicable Numbers"
] | [
"Quasiamicable Numbers/Examples/48,75",
"Definition:Quasiamicable Numbers",
"Definition:Quasiamicable Numbers",
"Definition:Abundant Number",
"Definition:Abundant Number"
] |
proofwiki-12955 | Bounds for Complex Logarithm | Let $\ln$ denote the complex logarithm.
Let $z \in \C$ with $\cmod z \le \dfrac 1 2$.
Then:
:$\dfrac 1 2 \cmod z \le \cmod {\map \ln {1 + z} } \le \dfrac 3 2 \cmod z$ | By definition of complex logarithm:
:$-\map \ln {1 + z} = \ds \sum_{n \mathop = 1}^\infty \frac {\paren {-z}^n} n$
Thus
{{begin-eqn}}
{{eqn | l = \cmod {1 - \frac {\map \ln {1 + z} } z}
| r = \cmod {\sum_{n \mathop = 2}^\infty \frac {\paren {-1}^n z^{n - 1} }n}
| c = Linear Combination of Convergent Seri... | Let $\ln$ denote the [[Definition:Complex Logarithm|complex logarithm]].
Let $z \in \C$ with $\cmod z \le \dfrac 1 2$.
Then:
:$\dfrac 1 2 \cmod z \le \cmod {\map \ln {1 + z} } \le \dfrac 3 2 \cmod z$ | By definition of [[Definition:Complex Logarithm|complex logarithm]]:
:$-\map \ln {1 + z} = \ds \sum_{n \mathop = 1}^\infty \frac {\paren {-z}^n} n$
Thus
{{begin-eqn}}
{{eqn | l = \cmod {1 - \frac {\map \ln {1 + z} } z}
| r = \cmod {\sum_{n \mathop = 2}^\infty \frac {\paren {-1}^n z^{n - 1} }n}
| c = [[... | Bounds for Complex Logarithm | https://proofwiki.org/wiki/Bounds_for_Complex_Logarithm | https://proofwiki.org/wiki/Bounds_for_Complex_Logarithm | [
"Logarithms"
] | [
"Definition:Natural Logarithm/Complex"
] | [
"Definition:Natural Logarithm/Complex",
"Linear Combination of Convergent Series",
"Triangle Inequality for Series",
"Sum of Infinite Geometric Sequence",
"Triangle Inequality"
] |
proofwiki-12956 | Reciprocal of Absolutely Convergent Product is Absolutely Convergent | Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a valued field.
Let $\sequence {1 + a_n}$ be a sequence of nonzero elements of $\mathbb K$.
Let the infinite product $\ds \prod_{n \mathop = 1}^\infty \left({1 + a_n}\right)$ converge absolutely to $a \in \mathbb K \setminus \set 0$.
Then $\ds \prod_{n \mathop = 1}^\infty ... | By continuity of $x \mapsto 1 / x$:
:$\ds \lim_{N \mathop \to \infty} \prod_{n \mathop = 1}^N \frac 1 {1 + a_n} = \frac 1 a$
It remains to prove the absolute convergence.
Because $\ds \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ converges absolutely, $\ds \sum_{n \mathop = 1}^\infty a_n$ converges absolutely.
By Fac... | Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a [[Definition:Valued Field|valued field]].
Let $\sequence {1 + a_n}$ be a [[Definition:Sequence|sequence]] of nonzero elements of $\mathbb K$.
Let the [[Definition:Infinite Product|infinite product]] $\ds \prod_{n \mathop = 1}^\infty \left({1 + a_n}\right)$ [[Definition... | By continuity of $x \mapsto 1 / x$:
:$\ds \lim_{N \mathop \to \infty} \prod_{n \mathop = 1}^N \frac 1 {1 + a_n} = \frac 1 a$
It remains to prove the absolute convergence.
Because $\ds \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ [[Definition:Absolute Convergence of Product|converges absolutely]], $\ds \sum_{n \mat... | Reciprocal of Absolutely Convergent Product is Absolutely Convergent | https://proofwiki.org/wiki/Reciprocal_of_Absolutely_Convergent_Product_is_Absolutely_Convergent | https://proofwiki.org/wiki/Reciprocal_of_Absolutely_Convergent_Product_is_Absolutely_Convergent | [
"Infinite Products"
] | [
"Definition:Valued Field",
"Definition:Sequence",
"Definition:Continued Product/Infinite",
"Definition:Absolute Convergence of Product",
"Definition:Absolute Convergence of Product"
] | [
"Definition:Absolute Convergence of Product",
"Definition:Absolutely Convergent Series",
"Factors in Absolutely Convergent Product Converge to One",
"Definition:Sufficiently Large",
"Definition:Sufficiently Large",
"Comparison Test",
"Definition:Absolutely Convergent Series",
"Definition:Absolute Conv... |
proofwiki-12957 | Equivalence of Definitions of Quasiamicable Numbers | Let $m \in \Z_{>0}$ and $n \in \Z_{>0}$ be (strictly) positive integers.
{{TFAE|def = Quasiamicable Numbers}} | Let $\map s n$ denote the sum of the proper divisors of (strictly) positive integer $n$.
The sum of all the divisors of a (strictly) positive integer $n$ is $\map {\sigma_1} n$, where $\sigma_1$ is the divisor sum function.
The proper divisors of $n$ are the divisors $n$ with $1$ and $n$ excluded.
Thus:
:$\map s n = \m... | Let $m \in \Z_{>0}$ and $n \in \Z_{>0}$ be [[Definition:Strictly Positive Integer|(strictly) positive integers]].
{{TFAE|def = Quasiamicable Numbers}} | Let $\map s n$ denote the [[Definition:Integer Addition|sum]] of the [[Definition:Proper Divisor of Integer|proper divisors]] of [[Definition:Strictly Positive Integer|(strictly) positive integer]] $n$.
The [[Definition:Integer Addition|sum]] of all the [[Definition:Divisor of Integer|divisors]] of a [[Definition:Stri... | Equivalence of Definitions of Quasiamicable Numbers | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Quasiamicable_Numbers | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Quasiamicable_Numbers | [
"Quasiamicable Numbers"
] | [
"Definition:Strictly Positive/Integer"
] | [
"Definition:Addition/Integers",
"Definition:Proper Divisor/Integer",
"Definition:Strictly Positive/Integer",
"Definition:Addition/Integers",
"Definition:Divisor (Algebra)/Integer",
"Definition:Strictly Positive/Integer",
"Definition:Divisor Sum Function",
"Definition:Proper Divisor/Integer",
"Defini... |
proofwiki-12958 | Equivalence of Definitions of Absolute Convergence of Product | Let $\struct {\mathbb K, \norm {\,\cdot\,} }$ be a valued field.
Let $\sequence {a_n}$ be a sequence in $\mathbb K$.
{{TFAE|def = Absolute Convergence of Product}}
=== Definition 1 ===
{{:Definition:Absolute Convergence of Product/General Definition/Definition 1}}
=== Definition 2 ===
{{:Definition:Absolute Convergence... | === 1 implies 2 ===
By the Monotone Convergence Theorem, it suffices to show that the partial sums of $\ds \sum_{n \mathop = 1}^\infty a_n$ are bounded.
Because $\ds \prod_{n \mathop = 1}^\infty \paren {1 + \norm {a_n} }$ converges, its partial products are bounded.
By Bounds for Finite Product of Real Numbers:
:$\ds \... | Let $\struct {\mathbb K, \norm {\,\cdot\,} }$ be a [[Definition:Valued Field|valued field]].
Let $\sequence {a_n}$ be a [[Definition:Sequence|sequence]] in $\mathbb K$.
{{TFAE|def = Absolute Convergence of Product}}
=== [[Definition:Absolute Convergence of Product/General Definition/Definition 1|Definition 1]] ===
{... | === 1 implies 2 ===
By the [[Monotone Convergence Theorem (Real Analysis)|Monotone Convergence Theorem]], it suffices to show that the [[Definition:Partial Sum|partial sums]] of $\ds \sum_{n \mathop = 1}^\infty a_n$ are bounded.
Because $\ds \prod_{n \mathop = 1}^\infty \paren {1 + \norm {a_n} }$ converges, its [[Def... | Equivalence of Definitions of Absolute Convergence of Product | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Absolute_Convergence_of_Product | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Absolute_Convergence_of_Product | [
"Infinite Products"
] | [
"Definition:Valued Field",
"Definition:Sequence",
"Definition:Absolute Convergence of Product/General Definition/Definition 1",
"Definition:Absolute Convergence of Product/General Definition/Definition 2"
] | [
"Monotone Convergence Theorem (Real Analysis)",
"Definition:Series/Sequence of Partial Sums",
"Definition:Sequence of Partial Products",
"Bounds for Finite Product of Real Numbers",
"Monotone Convergence Theorem (Real Analysis)",
"Definition:Sequence of Partial Products",
"Bounds for Finite Product of R... |
proofwiki-12959 | Factors in Convergent Product Converge to One | Let $\struct {\mathbb K, \norm {\,\cdot\,} }$ be a valued field.
Let the infinite product $\ds \prod_{n \mathop = 1}^\infty a_n$ be convergent.
Then $a_n \to 1$. | By definition of convergent product, there exists $n_0 \in \N$ such that:
:$a_n \ne 0$ for $n \ge n_0$
:the sequence of partial products of $\ds \prod_{n \mathop = n_0}^\infty a_n$ has a nonzero limit.
Let $p_n$ denote the $n$th partial product of $\ds \prod_{n \mathop = n_0}^\infty a_n$.
For $n > n_0$:
:$a_n = \dfrac ... | Let $\struct {\mathbb K, \norm {\,\cdot\,} }$ be a [[Definition:Valued Field|valued field]].
Let the [[Definition:Infinite Product|infinite product]] $\ds \prod_{n \mathop = 1}^\infty a_n$ be [[Definition:Convergent Product|convergent]].
Then $a_n \to 1$. | By definition of [[Definition:Convergent Product|convergent product]], there exists $n_0 \in \N$ such that:
:$a_n \ne 0$ for $n \ge n_0$
:the [[Definition:Sequence of Partial Products|sequence of partial products]] of $\ds \prod_{n \mathop = n_0}^\infty a_n$ has a nonzero [[Definition:Convergent Sequence|limit]].
Let ... | Factors in Convergent Product Converge to One | https://proofwiki.org/wiki/Factors_in_Convergent_Product_Converge_to_One | https://proofwiki.org/wiki/Factors_in_Convergent_Product_Converge_to_One | [
"Convergence",
"Infinite Products"
] | [
"Definition:Valued Field",
"Definition:Continued Product/Infinite",
"Definition:Convergent Product"
] | [
"Definition:Convergent Product",
"Definition:Sequence of Partial Products",
"Definition:Convergent Sequence",
"Definition:Sequence of Partial Products",
"Combination Theorem for Sequences"
] |
proofwiki-12960 | Logarithm of Convergent Product of Real Numbers | {{TFAE}}
:$(1): \quad$ The infinite product $\ds \prod_{n \mathop = 1}^\infty a_n$ converges to $a \in \R_{\ne 0}$.
:$(2): \quad$ The series $\ds \sum_{n \mathop = 1}^\infty \ln a_n$ converges to $\ln a$. | Let $p_n$ denote the $n$th partial product of $\ds \prod_{n \mathop = 1}^\infty a_n$.
Let $s_n$ denote the $n$th partial sum of $\ds \sum_{n \mathop = 1}^\infty \ln a_n$.
By Sum of Logarithms, $s_n = \map \ln {p_n}$. | {{TFAE}}
:$(1): \quad$ The [[Definition:Infinite Product|infinite product]] $\ds \prod_{n \mathop = 1}^\infty a_n$ [[Definition:Convergent Product|converges]] to $a \in \R_{\ne 0}$.
:$(2): \quad$ The [[Definition:Series|series]] $\ds \sum_{n \mathop = 1}^\infty \ln a_n$ [[Definition:Convergent Series of Numbers|converg... | Let $p_n$ denote the $n$th [[Definition:Partial Product|partial product]] of $\ds \prod_{n \mathop = 1}^\infty a_n$.
Let $s_n$ denote the $n$th [[Definition:Partial Sum|partial sum]] of $\ds \sum_{n \mathop = 1}^\infty \ln a_n$.
By [[Sum of Logarithms]], $s_n = \map \ln {p_n}$. | Logarithm of Convergent Product of Real Numbers | https://proofwiki.org/wiki/Logarithm_of_Convergent_Product_of_Real_Numbers | https://proofwiki.org/wiki/Logarithm_of_Convergent_Product_of_Real_Numbers | [
"Infinite Products",
"Convergence"
] | [
"Definition:Continued Product/Infinite",
"Definition:Convergent Product",
"Definition:Series",
"Definition:Convergent Series/Number Field"
] | [
"Definition:Sequence of Partial Products",
"Definition:Series/Sequence of Partial Sums",
"Sum of Logarithms"
] |
proofwiki-12961 | Factors in Absolutely Convergent Product Converge to One | Let $\struct {\mathbb K, \norm {\, \cdot \,} }$ be a valued field.
Let the infinite product $\ds \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ be absolutely convergent.
Then:
:$a_n \to 0$ | We have that $\ds \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ is absolutely convergent.
By the definition of absolutely convergent product, $\ds \prod_{n \mathop = 1}^\infty \paren {1 + \size {a_n} }$ is convergent.
{{begin-eqn}}
{{eqn | l = \prod_{n \mathop = 1}^\infty \paren {1 + \size {a_n} }
| r = \paren {... | Let $\struct {\mathbb K, \norm {\, \cdot \,} }$ be a [[Definition:Valued Field|valued field]].
Let the [[Definition:Infinite Product|infinite product]] $\ds \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ be [[Definition:Absolutely Convergent Product|absolutely convergent]].
Then:
:$a_n \to 0$ | We have that $\ds \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ is [[Definition:Absolutely Convergent Product|absolutely convergent]].
By the definition of [[Definition:Absolutely Convergent Product|absolutely convergent product]], $\ds \prod_{n \mathop = 1}^\infty \paren {1 + \size {a_n} }$ is [[Definition:Convergen... | Factors in Absolutely Convergent Product Converge to One/Proof 1 | https://proofwiki.org/wiki/Factors_in_Absolutely_Convergent_Product_Converge_to_One | https://proofwiki.org/wiki/Factors_in_Absolutely_Convergent_Product_Converge_to_One/Proof_1 | [
"Convergence",
"Infinite Products",
"Factors in Absolutely Convergent Product Converge to One"
] | [
"Definition:Valued Field",
"Definition:Continued Product/Infinite",
"Definition:Absolute Convergence of Product"
] | [
"Definition:Absolute Convergence of Product",
"Definition:Absolute Convergence of Product",
"Definition:Convergent Product",
"Definition:Multiplication",
"Definition:Convergent Product",
"Definition:Convergent Product",
"Definition:Absolutely Convergent Series",
"Terms in Convergent Series Converge to... |
proofwiki-12962 | Factors in Absolutely Convergent Product Converge to One | Let $\struct {\mathbb K, \norm {\, \cdot \,} }$ be a valued field.
Let the infinite product $\ds \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ be absolutely convergent.
Then:
:$a_n \to 0$ | We have that $\ds \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ is absolutely convergent.
Let $b_n = \paren {1 + a_n}$
Then $\ds \prod_{n \mathop = 1}^\infty b_n$ is absolutely convergent.
From Absolutely Convergent Product is Convergent, $\ds \prod_{n \mathop = 1}^\infty b_n$ is convergent.
By Factors in Convergent P... | Let $\struct {\mathbb K, \norm {\, \cdot \,} }$ be a [[Definition:Valued Field|valued field]].
Let the [[Definition:Infinite Product|infinite product]] $\ds \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ be [[Definition:Absolutely Convergent Product|absolutely convergent]].
Then:
:$a_n \to 0$ | We have that $\ds \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ is [[Definition:Absolutely Convergent Product|absolutely convergent]].
Let $b_n = \paren {1 + a_n}$
Then $\ds \prod_{n \mathop = 1}^\infty b_n$ is [[Definition:Absolutely Convergent Product|absolutely convergent]].
From [[Absolutely Convergent Product ... | Factors in Absolutely Convergent Product Converge to One/Proof 2 | https://proofwiki.org/wiki/Factors_in_Absolutely_Convergent_Product_Converge_to_One | https://proofwiki.org/wiki/Factors_in_Absolutely_Convergent_Product_Converge_to_One/Proof_2 | [
"Convergence",
"Infinite Products",
"Factors in Absolutely Convergent Product Converge to One"
] | [
"Definition:Valued Field",
"Definition:Continued Product/Infinite",
"Definition:Absolute Convergence of Product"
] | [
"Definition:Absolute Convergence of Product",
"Definition:Absolute Convergence of Product",
"Absolutely Convergent Product is Convergent",
"Definition:Convergent Product",
"Factors in Convergent Product Converge to One"
] |
proofwiki-12963 | Absolutely Convergent Product Does not Diverge to Zero | Let $\struct {\mathbb K, \norm {\, \cdot \,} }$ be a valued field.
Let the infinite product $\ds \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ be absolutely convergent.
Then it is not divergent to $0$. | By Factors in Absolutely Convergent Product Converge to One, $\norm {a_n} < 1$ for $n \ge n_0$.
Let $n_1 \ge n_0$.
{{AimForCont}} the product diverges to $0$.
Then:
:$\ds \prod_{n \mathop = n_1}^\infty \paren {1 + a_n} = 0$
By Norm of Limit:
:$\ds \prod_{n \mathop = n_1}^\infty \norm {1 + a_n} = 0$
By the Triangle Ineq... | Let $\struct {\mathbb K, \norm {\, \cdot \,} }$ be a [[Definition:Valued Field|valued field]].
Let the [[Definition:Infinite Product|infinite product]] $\ds \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ be [[Definition:Absolute Convergence of Product|absolutely convergent]].
Then it is not [[Definition:Divergent Pr... | By [[Factors in Absolutely Convergent Product Converge to One]], $\norm {a_n} < 1$ for $n \ge n_0$.
Let $n_1 \ge n_0$.
{{AimForCont}} the product [[Definition:Divergent Product|diverges]] to $0$.
Then:
:$\ds \prod_{n \mathop = n_1}^\infty \paren {1 + a_n} = 0$
By [[Norm of Limit]]:
:$\ds \prod_{n \mathop = n_1}^\in... | Absolutely Convergent Product Does not Diverge to Zero/Proof 1 | https://proofwiki.org/wiki/Absolutely_Convergent_Product_Does_not_Diverge_to_Zero | https://proofwiki.org/wiki/Absolutely_Convergent_Product_Does_not_Diverge_to_Zero/Proof_1 | [
"Infinite Products",
"Absolutely Convergent Product Does not Diverge to Zero"
] | [
"Definition:Valued Field",
"Definition:Continued Product/Infinite",
"Definition:Absolute Convergence of Product",
"Definition:Divergent Product"
] | [
"Factors in Absolutely Convergent Product Converge to One",
"Definition:Divergent Product",
"Norm of Limit",
"Triangle Inequality",
"Squeeze Theorem",
"Weierstrass Product Inequality",
"Definition:Absolute Convergence of Product",
"Definition:Sufficiently Large",
"Definition:Contradiction"
] |
proofwiki-12964 | Absolutely Convergent Product Does not Diverge to Zero | Let $\struct {\mathbb K, \norm {\, \cdot \,} }$ be a valued field.
Let the infinite product $\ds \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ be absolutely convergent.
Then it is not divergent to $0$. | We have that $\ds \prod_{n \mathop = 1}^\infty \paren {1 - \norm {a_n} }$ is absolutely convergent.
By Factors in Absolutely Convergent Product Converge to One, $\norm {a_n} < 1$ for $n \ge n_0$.
Thus $\ds \sum_{n \mathop = n_0}^\infty \map \ln {1 - \norm {a_n} }$ is absolutely convergent.
{{AimForCont}} the product di... | Let $\struct {\mathbb K, \norm {\, \cdot \,} }$ be a [[Definition:Valued Field|valued field]].
Let the [[Definition:Infinite Product|infinite product]] $\ds \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ be [[Definition:Absolute Convergence of Product|absolutely convergent]].
Then it is not [[Definition:Divergent Pr... | We have that $\ds \prod_{n \mathop = 1}^\infty \paren {1 - \norm {a_n} }$ is [[Definition:Absolute Convergence of Product|absolutely convergent]].
By [[Factors in Absolutely Convergent Product Converge to One]], $\norm {a_n} < 1$ for $n \ge n_0$.
Thus $\ds \sum_{n \mathop = n_0}^\infty \map \ln {1 - \norm {a_n} }$ is... | Absolutely Convergent Product Does not Diverge to Zero/Proof 2 | https://proofwiki.org/wiki/Absolutely_Convergent_Product_Does_not_Diverge_to_Zero | https://proofwiki.org/wiki/Absolutely_Convergent_Product_Does_not_Diverge_to_Zero/Proof_2 | [
"Infinite Products",
"Absolutely Convergent Product Does not Diverge to Zero"
] | [
"Definition:Valued Field",
"Definition:Continued Product/Infinite",
"Definition:Absolute Convergence of Product",
"Definition:Divergent Product"
] | [
"Definition:Absolute Convergence of Product",
"Factors in Absolutely Convergent Product Converge to One",
"Definition:Absolutely Convergent Series",
"Definition:Divergent Product",
"Norm of Limit",
"Triangle Inequality",
"Squeeze Theorem",
"Logarithm of Infinite Product of Real Numbers",
"Definition... |
proofwiki-12965 | Absolute Value of Absolutely Convergent Product is Absolutely Convergent | Let the infinite product $\ds \prod_{n \mathop = 1}^\infty a_n$ converge absolutely to $a \in \mathbb K$.
Then $\ds \prod_{n \mathop = 1}^\infty \norm {a_n}$ converges absolutely to $\norm a$. | By definition of absolute convergence of $\ds \prod_{n \mathop = 1}^\infty a_n$, $\ds \sum_{n \mathop = 1}^\infty \paren {a_n - 1}$ converges absolutely.
By the Triangle Inequality:
:$\size {\norm {a_n} - 1} \le \norm {a_n - 1}$
By the Comparison Test, $\ds \prod_{n \mathop = 1}^\infty \norm {a_n}$ converges absolutely... | Let the [[Definition:Infinite Product|infinite product]] $\ds \prod_{n \mathop = 1}^\infty a_n$ [[Definition:Absolute Convergence of Product|converge absolutely]] to $a \in \mathbb K$.
Then $\ds \prod_{n \mathop = 1}^\infty \norm {a_n}$ [[Definition:Absolute Convergence of Product|converges absolutely]] to $\norm a$. | By definition of [[Definition:Absolute Convergence of Product/General Definition/Definition 2|absolute convergence]] of $\ds \prod_{n \mathop = 1}^\infty a_n$, $\ds \sum_{n \mathop = 1}^\infty \paren {a_n - 1}$ [[Definition:Absolutely Convergent Series|converges absolutely]].
By the [[Triangle Inequality]]:
:$\size {\... | Absolute Value of Absolutely Convergent Product is Absolutely Convergent | https://proofwiki.org/wiki/Absolute_Value_of_Absolutely_Convergent_Product_is_Absolutely_Convergent | https://proofwiki.org/wiki/Absolute_Value_of_Absolutely_Convergent_Product_is_Absolutely_Convergent | [
"Infinite Products"
] | [
"Definition:Continued Product/Infinite",
"Definition:Absolute Convergence of Product",
"Definition:Absolute Convergence of Product"
] | [
"Definition:Absolute Convergence of Product/General Definition/Definition 2",
"Definition:Absolutely Convergent Series",
"Triangle Inequality",
"Comparison Test",
"Definition:Absolute Convergence of Product",
"Absolute Value is Continuous"
] |
proofwiki-12966 | Absolute Value of Convergent Infinite Product | Let the infinite product $\ds \prod_{n \mathop = 1}^\infty a_n$ converge to $a \in \mathbb K$.
Then $\ds \prod_{n \mathop = 1}^\infty \norm {a_n}$ converges to $\norm{a}$. | By Absolute Value of Limit of Sequence:
:$\ds \prod_{n \mathop = 1}^\infty \norm {a_n} = \norm a$
It remains to show that the product converges.
By the convergence, there exists $n_0 \in \N$ such that $a_n \ne 0$ for $n \ge n_0$.
Then $\norm {a_n} \ne 0$ for $n \ge n_0$.
Let $P_n$ denote the $n$th partial product of $\... | Let the [[Definition:Infinite Product|infinite product]] $\ds \prod_{n \mathop = 1}^\infty a_n$ [[Definition:Convergent Product|converge]] to $a \in \mathbb K$.
Then $\ds \prod_{n \mathop = 1}^\infty \norm {a_n}$ [[Definition:Convergent Product|converges]] to $\norm{a}$. | By [[Absolute Value of Limit of Sequence]]:
:$\ds \prod_{n \mathop = 1}^\infty \norm {a_n} = \norm a$
It remains to show that the product [[Definition:Convergent Product|converges]].
By the [[Definition:Convergent Product|convergence]], there exists $n_0 \in \N$ such that $a_n \ne 0$ for $n \ge n_0$.
Then $\norm {a_... | Absolute Value of Convergent Infinite Product | https://proofwiki.org/wiki/Absolute_Value_of_Convergent_Infinite_Product | https://proofwiki.org/wiki/Absolute_Value_of_Convergent_Infinite_Product | [
"Infinite Products"
] | [
"Definition:Continued Product/Infinite",
"Definition:Convergent Product",
"Definition:Convergent Product"
] | [
"Absolute Value of Limit of Sequence",
"Definition:Convergent Product",
"Definition:Convergent Product",
"Definition:Sequence of Partial Products",
"Definition:Sequence of Partial Products",
"Convergence of Absolute Value of Sequence",
"Definition:Convergent Product"
] |
proofwiki-12967 | Absolute Value of Divergent Infinite Product | The infinite product $\ds \prod_{n \mathop = 1}^\infty a_n$ diverges to $0$ {{iff}} $\ds \prod_{n \mathop = 1}^\infty \norm {a_n}$ diverges to $0$. | === 1 implies 2 === | The [[Definition:Infinite Product|infinite product]] $\ds \prod_{n \mathop = 1}^\infty a_n$ [[Definition:Divergent Product|diverges]] to $0$ {{iff}} $\ds \prod_{n \mathop = 1}^\infty \norm {a_n}$ [[Definition:Divergent Product|diverges]] to $0$. | === 1 implies 2 === | Absolute Value of Divergent Infinite Product | https://proofwiki.org/wiki/Absolute_Value_of_Divergent_Infinite_Product | https://proofwiki.org/wiki/Absolute_Value_of_Divergent_Infinite_Product | [
"Infinite Products"
] | [
"Definition:Continued Product/Infinite",
"Definition:Divergent Product",
"Definition:Divergent Product"
] | [] |
proofwiki-12968 | Logarithm of Divergent Product of Real Numbers/Infinity | {{TFAE}}:
:$(1): \quad$ The infinite product $\ds \prod_{n \mathop = 1}^\infty a_n$ diverges to $+\infty$.
:$(2): \quad$ The series $\ds \sum_{n \mathop = 1}^\infty \log a_n$ diverges to $+\infty$. | Let $p_n$ denote the $n$th partial product of $\ds \prod_{n \mathop = 1}^\infty a_n$.
Let $s_n$ denote the $n$th partial sum of $\ds \sum_{n \mathop = 1}^\infty\log a_n$.
By Sum of Logarithms:
:$s_n = \map \log {p_n}$
{{ProofWanted}} | {{TFAE}}:
:$(1): \quad$ The [[Definition:Infinite Product|infinite product]] $\ds \prod_{n \mathop = 1}^\infty a_n$ [[Definition:Divergent Product|diverges]] to $+\infty$.
:$(2): \quad$ The [[Definition:Series|series]] $\ds \sum_{n \mathop = 1}^\infty \log a_n$ [[Definition:Divergent Series|diverges]] to $+\infty$. | Let $p_n$ denote the $n$th [[Definition:Partial Product|partial product]] of $\ds \prod_{n \mathop = 1}^\infty a_n$.
Let $s_n$ denote the $n$th [[Definition:Partial Sum|partial sum]] of $\ds \sum_{n \mathop = 1}^\infty\log a_n$.
By [[Sum of Logarithms]]:
:$s_n = \map \log {p_n}$
{{ProofWanted}} | Logarithm of Divergent Product of Real Numbers/Infinity | https://proofwiki.org/wiki/Logarithm_of_Divergent_Product_of_Real_Numbers/Infinity | https://proofwiki.org/wiki/Logarithm_of_Divergent_Product_of_Real_Numbers/Infinity | [
"Logarithm of Divergent Product of Real Numbers"
] | [
"Definition:Continued Product/Infinite",
"Definition:Divergent Product",
"Definition:Series",
"Definition:Divergent Series"
] | [
"Definition:Sequence of Partial Products",
"Definition:Series/Sequence of Partial Sums",
"Sum of Logarithms"
] |
proofwiki-12969 | Logarithm of Divergent Product of Real Numbers/Zero | {{TFAE}}
{{begin-itemize}}
{{item|*|The infinite product $\ds \prod_{n \mathop {{=}} 1}^\infty a_n$ diverges to $0$.}}
{{item|*|The series $\ds \sum_{n \mathop {{=}} 1}^\infty \log a_n$ diverges to $-\infty$.}}
{{end-itemize}} | Let $p_n$ denote the $n$th partial product of $\ds \prod_{n \mathop = 1}^\infty a_n$.
Let $s_n$ denote the $n$th partial sum of $\ds \sum_{n \mathop = 1}^\infty \log a_n$.
By Sum of Logarithms:
:$s_n = \log p_n$
{{ProofWanted}} | {{TFAE}}
{{begin-itemize}}
{{item|*|The [[Definition:Infinite Product|infinite product]] $\ds \prod_{n \mathop {{=}} 1}^\infty a_n$ [[Definition:Divergence to Zero|diverges]] to $0$.}}
{{item|*|The [[Definition:Series|series]] $\ds \sum_{n \mathop {{=}} 1}^\infty \log a_n$ [[Definition:Divergent Series|diverges]] to $-... | Let $p_n$ denote the $n$th [[Definition:Partial Product|partial product]] of $\ds \prod_{n \mathop = 1}^\infty a_n$.
Let $s_n$ denote the $n$th [[Definition:Partial Sum|partial sum]] of $\ds \sum_{n \mathop = 1}^\infty \log a_n$.
By [[Sum of Logarithms]]:
:$s_n = \log p_n$
{{ProofWanted}} | Logarithm of Divergent Product of Real Numbers/Zero | https://proofwiki.org/wiki/Logarithm_of_Divergent_Product_of_Real_Numbers/Zero | https://proofwiki.org/wiki/Logarithm_of_Divergent_Product_of_Real_Numbers/Zero | [
"Logarithm of Divergent Product of Real Numbers"
] | [
"Definition:Continued Product/Infinite",
"Definition:Divergent Product/Divergence to Zero",
"Definition:Series",
"Definition:Divergent Series"
] | [
"Definition:Sequence of Partial Products",
"Definition:Series/Sequence of Partial Sums",
"Sum of Logarithms"
] |
proofwiki-12970 | Product of Convergent Products is Convergent | Let $\struct {\mathbb K, \norm {\,\cdot\,} }$ be a valued field.
Let $\ds \prod_{n \mathop = 1}^\infty a_n$ converge to $a$.
Let $\ds \prod_{n \mathop = 1}^\infty b_n$ converge to $b$.
Then $\ds \prod_{n \mathop = 1}^\infty a_n b_n$ converges to $ab$. | Let $n_0 \in \N$ such that $a_n \ne 0$ for $n> n_0$.
Let $n_1 \in \N$ such that $b_n \ne 0$ for $n> n_1$.
Then $a_n b_n \ne 0$ for $n > n_2 = \max \set {n_0, n_1}$.
Let $p_n$ be the $n$th partial product of $\ds \prod_{n \mathop = n_2 + 1}^\infty a_n$.
Let $q_n$ be the $n$th partial product of $\ds \prod_{n \mathop = n... | Let $\struct {\mathbb K, \norm {\,\cdot\,} }$ be a [[Definition:Valued Field|valued field]].
Let $\ds \prod_{n \mathop = 1}^\infty a_n$ [[Definition:Convergent Product|converge]] to $a$.
Let $\ds \prod_{n \mathop = 1}^\infty b_n$ [[Definition:Convergent Product|converge]] to $b$.
Then $\ds \prod_{n \mathop = 1}^\in... | Let $n_0 \in \N$ such that $a_n \ne 0$ for $n> n_0$.
Let $n_1 \in \N$ such that $b_n \ne 0$ for $n> n_1$.
Then $a_n b_n \ne 0$ for $n > n_2 = \max \set {n_0, n_1}$.
Let $p_n$ be the $n$th [[Definition:Partial Product|partial product]] of $\ds \prod_{n \mathop = n_2 + 1}^\infty a_n$.
Let $q_n$ be the $n$th [[Definit... | Product of Convergent Products is Convergent | https://proofwiki.org/wiki/Product_of_Convergent_Products_is_Convergent | https://proofwiki.org/wiki/Product_of_Convergent_Products_is_Convergent | [
"Infinite Products"
] | [
"Definition:Valued Field",
"Definition:Convergent Product",
"Definition:Convergent Product",
"Definition:Convergent Product"
] | [
"Definition:Sequence of Partial Products",
"Definition:Sequence of Partial Products",
"Definition:Sequence of Partial Products",
"Definition:Convergent Sequence/Metric Space",
"Definition:Convergent Product",
"Definition:Sequence of Partial Products",
"Definition:Sequence of Partial Products",
"Defini... |
proofwiki-12971 | Trimorphic Number is not necessarily Automorphic | Let $n \in \Z_{>0}$ be a trimorphic number.
Then it is not necessarily the case that $n$ is also an automorphic number. | Take as an example $n = 49$.
We have that:
:$49^3 = 117 \, 6 \mathbf{49}$
demonstrating that $49$ is trimorphic.
However, we also have that:
:$49^2 = 2401$
demonstrating that $49$ is not automorphic.
{{qed}} | Let $n \in \Z_{>0}$ be a [[Definition:Trimorphic Number|trimorphic number]].
Then it is not necessarily the case that $n$ is also an [[Definition:Automorphic Number|automorphic number]]. | Take as an example $n = 49$.
We have that:
:$49^3 = 117 \, 6 \mathbf{49}$
demonstrating that $49$ is [[Definition:Trimorphic Number|trimorphic]].
However, we also have that:
:$49^2 = 2401$
demonstrating that $49$ is not [[Definition:Automorphic Number|automorphic]].
{{qed}} | Trimorphic Number is not necessarily Automorphic | https://proofwiki.org/wiki/Trimorphic_Number_is_not_necessarily_Automorphic | https://proofwiki.org/wiki/Trimorphic_Number_is_not_necessarily_Automorphic | [
"Trimorphic Numbers",
"Automorphic Numbers"
] | [
"Definition:Trimorphic Number",
"Definition:Automorphic Number"
] | [
"Definition:Trimorphic Number",
"Definition:Automorphic Number"
] |
proofwiki-12972 | Reciprocal of 49 shows Powers of 2 in Decimal Expansion | The decimal expansion of the reciprocal of $49$ contains the powers of $2$:
:$\dfrac 1 {49} = 0 \cdotp \dot 02040 \, 81632 \, 65306 \, 12244 \, 89795 \, 91836 \, 73469 \, 38775 \, 5 \dot 1$ | From Reciprocal of $49$:
{{:Reciprocal of 49}}
Adding up powers of $2$, shifted appropriately to the right:
<pre>
02
04
08
16
32
64
128
256
512
1024
2048
4096
8192
... | The [[Definition:Decimal Expansion|decimal expansion]] of the [[Definition:Reciprocal|reciprocal]] of $49$ contains the [[Definition:Integer Power|powers]] of $2$:
:$\dfrac 1 {49} = 0 \cdotp \dot 02040 \, 81632 \, 65306 \, 12244 \, 89795 \, 91836 \, 73469 \, 38775 \, 5 \dot 1$ | From [[Reciprocal of 49|Reciprocal of $49$]]:
{{:Reciprocal of 49}}
Adding up powers of $2$, shifted appropriately to the right:
<pre>
02
04
08
16
32
64
128
256
512
1024
2048
4096
... | Reciprocal of 49 shows Powers of 2 in Decimal Expansion | https://proofwiki.org/wiki/Reciprocal_of_49_shows_Powers_of_2_in_Decimal_Expansion | https://proofwiki.org/wiki/Reciprocal_of_49_shows_Powers_of_2_in_Decimal_Expansion | [
"49",
"Examples of Reciprocals"
] | [
"Definition:Decimal Expansion",
"Definition:Reciprocal",
"Definition:Power (Algebra)/Integer"
] | [
"Reciprocal of 49",
"Definition:Power (Algebra)/Integer",
"Sum of Infinite Geometric Sequence/Corollary 1"
] |
proofwiki-12973 | Product of Absolutely Convergent Products is Absolutely Convergent | Let $\struct {\mathbb K, \norm {\,\cdot\,} }$ be a valued field.
Let $\ds \prod_{n \mathop = 1}^\infty a_n$ converge absolutely.
Let $\ds \prod_{n \mathop = 1}^\infty b_n$ converge absolutely.
Then $\ds \prod_{n \mathop = 1}^\infty a_nb_n$ converges absolutely. | We have:
:$a_n b_n - 1 = \paren {a_n - 1} \paren {b_n - 1} + \paren {a_n - 1} + \paren {b_n - 1}$
By the Triangle Inequality:
:$\norm {a_n b_n - 1} \le \norm {a_n - 1} \norm {b_n - 1} + \norm {a_n - 1} + \norm {b_n -1}$
By the absolute convergence, $\ds \sum_{n \mathop = 1}^\infty \norm {a_n - 1}$ and $\ds \sum_{n \mat... | Let $\struct {\mathbb K, \norm {\,\cdot\,} }$ be a [[Definition:Valued Field|valued field]].
Let $\ds \prod_{n \mathop = 1}^\infty a_n$ [[Definition:Absolutely Convergent Product|converge absolutely]].
Let $\ds \prod_{n \mathop = 1}^\infty b_n$ [[Definition:Absolutely Convergent Product|converge absolutely]].
Then ... | We have:
:$a_n b_n - 1 = \paren {a_n - 1} \paren {b_n - 1} + \paren {a_n - 1} + \paren {b_n - 1}$
By the [[Triangle Inequality]]:
:$\norm {a_n b_n - 1} \le \norm {a_n - 1} \norm {b_n - 1} + \norm {a_n - 1} + \norm {b_n -1}$
By the [[Definition:Absolutely Convergent Product|absolute convergence]], $\ds \sum_{n \mathop... | Product of Absolutely Convergent Products is Absolutely Convergent | https://proofwiki.org/wiki/Product_of_Absolutely_Convergent_Products_is_Absolutely_Convergent | https://proofwiki.org/wiki/Product_of_Absolutely_Convergent_Products_is_Absolutely_Convergent | [
"Infinite Products"
] | [
"Definition:Valued Field",
"Definition:Absolute Convergence of Product",
"Definition:Absolute Convergence of Product",
"Definition:Absolute Convergence of Product"
] | [
"Triangle Inequality",
"Definition:Absolute Convergence of Product",
"Definition:Convergent Series",
"Inner Product of Absolutely Convergent Series",
"Definition:Convergent Series",
"Comparison Test",
"Definition:Convergent Series"
] |
proofwiki-12974 | Convergent Product Satisfies Cauchy Criterion | Let $\struct {\mathbb K, \norm {\,\cdot\,} }$ be a valued field.
Let the infinite product $\ds \prod_{n \mathop = 1}^\infty a_n$ be convergent.
Then it satisfies Cauchy's criterion for products. | Let $\epsilon > 0$.
Let $n_0 \in \N$ be such that $\ds \prod_{n \mathop = n_0}^\infty a_n$ converges to some $a \in \mathbb K \setminus \set 0$.
By Convergent Sequence is Cauchy Sequence, there exists $N_0 \ge n_0$ such that:
:$\ds \norm {\prod_{n \mathop = n_0}^k a_n - \prod_{n \mathop = n_0}^l a_n} \le \epsilon$
for ... | Let $\struct {\mathbb K, \norm {\,\cdot\,} }$ be a [[Definition:Valued Field|valued field]].
Let the [[Definition:Infinite Product|infinite product]] $\ds \prod_{n \mathop = 1}^\infty a_n$ be [[Definition:Convergence of Product|convergent]].
Then it satisfies [[Definition:Cauchy's Criterion for Products|Cauchy's cri... | Let $\epsilon > 0$.
Let $n_0 \in \N$ be such that $\ds \prod_{n \mathop = n_0}^\infty a_n$ [[Definition:Convergence of Product|converges]] to some $a \in \mathbb K \setminus \set 0$.
By [[Convergent Sequence is Cauchy Sequence]], there exists $N_0 \ge n_0$ such that:
:$\ds \norm {\prod_{n \mathop = n_0}^k a_n - \prod... | Convergent Product Satisfies Cauchy Criterion | https://proofwiki.org/wiki/Convergent_Product_Satisfies_Cauchy_Criterion | https://proofwiki.org/wiki/Convergent_Product_Satisfies_Cauchy_Criterion | [
"Infinite Products",
"Cauchy Sequences"
] | [
"Definition:Valued Field",
"Definition:Continued Product/Infinite",
"Definition:Convergent Product",
"Definition:Cauchy's Criterion for Products"
] | [
"Definition:Convergent Product",
"Convergent Sequence is Cauchy Sequence",
"Sequence Converges to Within Half Limit"
] |
proofwiki-12975 | Uniform Product of Continuous Functions is Continuous | Let $X$ be a metric space.
Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a valued field.
Let $\sequence {f_n}$ be a sequence of bounded continuous mappings $f_n: X \to \mathbb K$.
Let the product $\ds \prod_{n \mathop = 1}^\infty f_n$ converge uniformly to $f$.
Then $f$ is continuous. | Let $n_0 \in \N$ be such that the sequence of partial products of $\ds \prod_{n \mathop = n_0}^\infty f_n$ converges uniformly.
By the Uniform Limit Theorem, $\ds \prod_{n \mathop = n_0}^\infty f_n$ is continuous.
Because $f_1, \ldots, f_{n_0 - 1}$ are continuous, so is $\ds \prod_{n \mathop = 1}^\infty f_n$.
{{qed}} | Let $X$ be a [[Definition:Metric Space|metric space]].
Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a [[Definition:Valued Field|valued field]].
Let $\sequence {f_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Bounded Mapping|bounded]] [[Definition:Continuous Mapping|continuous mappings]] $f_n: X \to \mat... | Let $n_0 \in \N$ be such that the [[Definition:Sequence of Partial Products|sequence of partial products]] of $\ds \prod_{n \mathop = n_0}^\infty f_n$ [[Definition:Uniform Convergence|converges uniformly]].
By the [[Uniform Limit Theorem]], $\ds \prod_{n \mathop = n_0}^\infty f_n$ is [[Definition:Continuous Mapping|co... | Uniform Product of Continuous Functions is Continuous/Proof 1 | https://proofwiki.org/wiki/Uniform_Product_of_Continuous_Functions_is_Continuous | https://proofwiki.org/wiki/Uniform_Product_of_Continuous_Functions_is_Continuous/Proof_1 | [
"Uniform Convergence",
"Infinite Products",
"Uniform Product of Continuous Functions is Continuous"
] | [
"Definition:Metric Space",
"Definition:Valued Field",
"Definition:Sequence",
"Definition:Bounded Mapping",
"Definition:Continuous Mapping",
"Definition:Uniform Convergence of Product",
"Definition:Continuous Mapping"
] | [
"Definition:Sequence of Partial Products",
"Definition:Uniform Convergence",
"Uniform Limit Theorem",
"Definition:Continuous Mapping"
] |
proofwiki-12976 | Logarithm of Infinite Product of Complex Functions | Let $X$ be a weakly locally compact topological space.
Let $\sequence {f_n}$ be a sequence of everywhere nonzero continuous mappings $f_n: X \to \C$.
{{TFAE}}
{{begin-itemize}}
{{item|(1):|The product $\ds \prod_{n \mathop {{=}} 1}^\infty f_n$ converges locally uniformly to $f$}}
{{item|(2):|The series $\ds \sum_{n \ma... | === 1 implies 2 ===
It suffices to show that:
{{begin-itemize}}
{{item|(3):|$\ds \sum_{n \mathop {{=}} 1}^\infty \map \Re {\ln f_n} {{=}} \map \Re {\ln f}$ locally uniformly}}
{{item|(4):|$\ds \sum_{n \mathop {{=}} 1}^\infty \map \Im {\ln f_n} {{=}} \map \Im {\ln f} + 2 k \pi$ locally uniformly for some $k: K \to \Z$}}... | Let $X$ be a [[Definition:Weakly Locally Compact Space|weakly locally compact topological space]].
Let $\sequence {f_n}$ be a [[Definition:Sequence|sequence]] of everywhere nonzero [[Definition:Continuous Mapping|continuous mappings]] $f_n: X \to \C$.
{{TFAE}}
{{begin-itemize}}
{{item|(1):|The product $\ds \prod_{n ... | === 1 implies 2 ===
It suffices to show that:
{{begin-itemize}}
{{item|(3):|$\ds \sum_{n \mathop {{=}} 1}^\infty \map \Re {\ln f_n} {{=}} \map \Re {\ln f}$ [[Definition:Locally Uniform Convergence|locally uniformly]]}}
{{item|(4):|$\ds \sum_{n \mathop {{=}} 1}^\infty \map \Im {\ln f_n} {{=}} \map \Im {\ln f} + 2 k \pi... | Logarithm of Infinite Product of Complex Functions | https://proofwiki.org/wiki/Logarithm_of_Infinite_Product_of_Complex_Functions | https://proofwiki.org/wiki/Logarithm_of_Infinite_Product_of_Complex_Functions | [
"Infinite Products"
] | [
"Definition:Weakly Locally Compact Space",
"Definition:Sequence",
"Definition:Continuous Mapping",
"Definition:Locally Uniform Convergence of Product",
"Definition:Locally Uniform Convergence",
"Definition:Mapping"
] | [
"Definition:Locally Uniform Convergence",
"Definition:Locally Uniform Convergence",
"Definition:Locally Uniform Convergence"
] |
proofwiki-12977 | Logarithmic Derivative of Infinite Product of Analytic Functions | Let $D \subseteq \C$ be open.
Let $\sequence {f_n}$ be a sequence of analytic functions $f_n: D \to \C$.
Let none of the $f_n$ be identically zero on any open subset of $D$.
Let the product $\ds \prod_{n \mathop = 1}^\infty f_n$ converge locally uniformly to $f$.
Then:
:$\ds \dfrac {f'} f = \sum_{n \mathop = 1}^\infty ... | {{questionable|this needs a stronger version of {{Corollary|Logarithm of Infinite Product of Complex Functions}}, where the logarithms are analytic}}
Note that by Infinite Product of Analytic Functions is Analytic, $f$ is analytic.
Let $z_0 \in D$ with $\map f {z_0} \ne 0$.
By {{Corollary|Logarithm of Infinite Product ... | Let $D \subseteq \C$ be [[Definition:Open Set (Complex Analysis)|open]].
Let $\sequence {f_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Analytic Function|analytic functions]] $f_n: D \to \C$.
Let none of the $f_n$ be identically zero on any [[Definition:Open Set (Complex Analysis)|open subset]] of $D$.
... | {{questionable|this needs a stronger version of {{Corollary|Logarithm of Infinite Product of Complex Functions}}, where the logarithms are analytic}}
Note that by [[Infinite Product of Analytic Functions is Analytic]], $f$ is [[Definition:Analytic Function|analytic]].
Let $z_0 \in D$ with $\map f {z_0} \ne 0$.
By {{C... | Logarithmic Derivative of Infinite Product of Analytic Functions | https://proofwiki.org/wiki/Logarithmic_Derivative_of_Infinite_Product_of_Analytic_Functions | https://proofwiki.org/wiki/Logarithmic_Derivative_of_Infinite_Product_of_Analytic_Functions | [
"Analytic Functions",
"Infinite Products",
"Logarithmic Differentiation"
] | [
"Definition:Open Set/Complex Analysis",
"Definition:Sequence",
"Definition:Analytic Function",
"Definition:Open Set/Complex Analysis",
"Definition:Locally Uniform Convergence of Product",
"Definition:Locally Uniform Convergence"
] | [
"Infinite Product of Analytic Functions is Analytic",
"Definition:Analytic Function",
"Definition:Open Neighborhood (Metric Space)",
"Definition:Uniform Convergence",
"Logarithmic Derivative of Product of Analytic Functions",
"Logarithmic Derivative is Derivative of Logarithm",
"Derivative of Uniform Li... |
proofwiki-12978 | Galois Connection implies Upper Adjoint is Surjection iff Lower Adjoint is Injection | Let $L = \struct {S, \preceq}$ and $R = \struct {T, \precsim}$ be ordered sets.
Let $g: S \to T$ and $d: T \to S$ be mappings such that $\struct {g, d}$ is a Galois connection.
Then $g$ is a surjection {{iff}} $d$ is an injection. | === Sufficient Condition ===
Assume that
:$d$ is a surjection.
By Upper Adjoint of Galois Connection is Surjection implies Lower Adjoint at Element is Minimum of Preimage of Singleton of Element
:$\forall t \in T: \map d t = \min \set {g^{-1} \sqbrk {\set t} }$
By Lower Adjoint at Element is Minimum of Preimage of Sing... | Let $L = \struct {S, \preceq}$ and $R = \struct {T, \precsim}$ be [[Definition:Ordered Set|ordered sets]].
Let $g: S \to T$ and $d: T \to S$ be [[Definition:Mapping|mappings]] such that $\struct {g, d}$ is a [[Definition:Galois Connection|Galois connection]].
Then $g$ is a [[Definition:Surjection|surjection]] {{iff}... | === Sufficient Condition ===
Assume that
:$d$ is a [[Definition:Surjection|surjection]].
By [[Upper Adjoint of Galois Connection is Surjection implies Lower Adjoint at Element is Minimum of Preimage of Singleton of Element]]
:$\forall t \in T: \map d t = \min \set {g^{-1} \sqbrk {\set t} }$
By [[Lower Adjoint at Ele... | Galois Connection implies Upper Adjoint is Surjection iff Lower Adjoint is Injection | https://proofwiki.org/wiki/Galois_Connection_implies_Upper_Adjoint_is_Surjection_iff_Lower_Adjoint_is_Injection | https://proofwiki.org/wiki/Galois_Connection_implies_Upper_Adjoint_is_Surjection_iff_Lower_Adjoint_is_Injection | [
"Galois Connections"
] | [
"Definition:Ordered Set",
"Definition:Mapping",
"Definition:Galois Connection",
"Definition:Surjection",
"Definition:Injection"
] | [
"Definition:Surjection",
"Upper Adjoint of Galois Connection is Surjection implies Lower Adjoint at Element is Minimum of Preimage of Singleton of Element",
"Lower Adjoint at Element is Minimum of Preimage of Singleton of Element implies Composition is Identity",
"Injection iff Left Inverse",
"Definition:In... |
proofwiki-12979 | Derivative of Uniform Limit of Analytic Functions | Let $U$ be an open subset of $\C$.
Let $\sequence {f_n}_{n \mathop \in \N}$ be a sequence of analytic functions $f_n : U \to \C$.
Let $\sequence {f_n}$ converge locally uniformly to $f$ on $U$.
Let $f'$ denote the derivative of $f$.
Then the sequence $\sequence { {f_n}'}_{n \mathop \in \N}$ converges locally uniformly ... | Let $a \in U$.
By definition of locally uniform convergence, there exists an open disk $\map {D_{2 r} } a \subseteq U$ such that $f_n$ converges uniformly to $f$ on $\map {D_{2 r} } a$.
That is:
:$\ds (1): \quad \lim_{n \mathop \to \infty} \sup_{z \mathop \in \map {D_{2 r} } a} \cmod {\map {f_n} z - \map f z} = 0$
We ... | Let $U$ be an [[Definition:Open Set (Complex Analysis)|open subset]] of $\C$.
Let $\sequence {f_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] of [[Definition:Analytic Function|analytic functions]] $f_n : U \to \C$.
Let $\sequence {f_n}$ [[Definition:Locally Uniform Convergence|converge locally uniforml... | Let $a \in U$.
By definition of [[Definition:Locally Uniform Convergence|locally uniform convergence]], there exists an [[Definition:Open Complex Disk|open disk]] $\map {D_{2 r} } a \subseteq U$ such that $f_n$ [[Definition:Uniform Convergence|converges uniformly]] to $f$ on $\map {D_{2 r} } a$.
That is:
:$\ds (1): \... | Derivative of Uniform Limit of Analytic Functions | https://proofwiki.org/wiki/Derivative_of_Uniform_Limit_of_Analytic_Functions | https://proofwiki.org/wiki/Derivative_of_Uniform_Limit_of_Analytic_Functions | [
"Complex Analysis",
"Complex Analytic Functions",
"Uniform Convergence"
] | [
"Definition:Open Set/Complex Analysis",
"Definition:Sequence",
"Definition:Analytic Function",
"Definition:Locally Uniform Convergence",
"Definition:Derivative",
"Definition:Sequence",
"Definition:Locally Uniform Convergence"
] | [
"Definition:Locally Uniform Convergence",
"Definition:Complex Disk/Open",
"Definition:Uniform Convergence",
"Triangle Inequality/Complex Numbers",
"Cauchy's Integral Formula/General Result",
"Estimation Lemma for Contour Integrals"
] |
proofwiki-12980 | Derivative of Infinite Product of Analytic Functions | Let $D \subset \C$ be open.
Let $\sequence {f_n}$ be a sequence of analytic functions $f_n: D \to \C$.
Let the product $\ds \prod_{n \mathop = 1}^\infty f_n$ converge locally uniformly to $f$.
Then:
:$\ds f' = \sum_{n \mathop = 1}^\infty f_n' \cdot \prod_{\substack {k \mathop = 1 \\ k \mathop \ne n} }^\infty f_k$
and t... | By Infinite Product of Analytic Functions is Analytic, $f$ is analytic.
We may suppose none of the $f_n$ is identically zero on any open subset of $D$.
Let $E = D \setminus \set {z \in D: \map f z = 0}$.
By Logarithmic Derivative of Infinite Product of Analytic Functions, $\ds \frac {f'} f = \sum_{n \mathop = 1}^\infty... | Let $D \subset \C$ be [[Definition:Open Set (Complex Analysis)|open]].
Let $\sequence {f_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Analytic Function|analytic functions]] $f_n: D \to \C$.
Let the [[Definition:Infinite Product|product]] $\ds \prod_{n \mathop = 1}^\infty f_n$ [[Definition:Locally Uniform... | By [[Infinite Product of Analytic Functions is Analytic]], $f$ is [[Definition:Analytic Function|analytic]].
We may suppose none of the $f_n$ is identically zero on any [[Definition:Open Set (Complex Analysis)|open subset]] of $D$.
Let $E = D \setminus \set {z \in D: \map f z = 0}$.
By [[Logarithmic Derivative of In... | Derivative of Infinite Product of Analytic Functions | https://proofwiki.org/wiki/Derivative_of_Infinite_Product_of_Analytic_Functions | https://proofwiki.org/wiki/Derivative_of_Infinite_Product_of_Analytic_Functions | [
"Complex Analysis",
"Infinite Products"
] | [
"Definition:Open Set/Complex Analysis",
"Definition:Sequence",
"Definition:Analytic Function",
"Definition:Continued Product/Infinite",
"Definition:Locally Uniform Convergence of Product",
"Definition:Series",
"Definition:Locally Uniform Convergence"
] | [
"Infinite Product of Analytic Functions is Analytic",
"Definition:Analytic Function",
"Definition:Open Set/Complex Analysis",
"Logarithmic Derivative of Infinite Product of Analytic Functions",
"Definition:Locally Uniform Convergence",
"Linear Combination of Convergent Series",
"Uniformly Convergent Seq... |
proofwiki-12981 | Uniformly Convergent Sequence Multiplied with Function | Let $X$ be a set.
Let $V$ be a normed vector space over $\mathbb K$.
Let $\sequence {f_n}$ be a sequence of mappings $f_n: X \to V$.
Let $\sequence {f_n}$ be uniformly convergent.
Let $g: X \to \mathbb K$ be bounded.
Then $\sequence {f_n g}$ is uniformly convergent. | Denote $\norm {\, \cdot \,}$ as the norm on $V$.
Let $\epsilon > 0$.
By boundedness of $g$:
:$\exists M \in \R: \forall x \in X: \norm {\map g x} < M$
By uniformly convergence of $\sequence {f_n}$:
:$\exists f: X \to \mathbb K: \exists N \in \R: \forall x \in X: \norm {\map {f_n} x - \map f x} < \dfrac \epsilon M$
Pick... | Let $X$ be a [[Definition:Set|set]].
Let $V$ be a [[Definition:Normed Vector Space|normed vector space]] over $\mathbb K$.
Let $\sequence {f_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Mapping|mappings]] $f_n: X \to V$.
Let $\sequence {f_n}$ be [[Definition:Uniformly Convergent|uniformly convergent]].
... | Denote $\norm {\, \cdot \,}$ as the [[Definition:Norm on Vector Space|norm]] on $V$.
Let $\epsilon > 0$.
By [[Definition:Bounded Mapping|boundedness]] of $g$:
:$\exists M \in \R: \forall x \in X: \norm {\map g x} < M$
By [[Definition:Uniformly Convergent|uniformly convergence]] of $\sequence {f_n}$:
:$\exists f: X... | Uniformly Convergent Sequence Multiplied with Function | https://proofwiki.org/wiki/Uniformly_Convergent_Sequence_Multiplied_with_Function | https://proofwiki.org/wiki/Uniformly_Convergent_Sequence_Multiplied_with_Function | [
"Uniform Convergence"
] | [
"Definition:Set",
"Definition:Normed Vector Space",
"Definition:Sequence",
"Definition:Mapping",
"Definition:Uniform Convergence",
"Definition:Bounded Mapping",
"Definition:Uniform Convergence"
] | [
"Definition:Norm/Vector Space",
"Definition:Bounded Mapping",
"Definition:Uniform Convergence",
"Definition:Uniform Convergence"
] |
proofwiki-12982 | Logarithmic Derivative of Product of Analytic Functions | Let $D \subset \C$ be open.
Let $f, g: D \to \C$ be analytic.
Let $f g$ be their pointwise product.
Let $z \in D$ with $\map f z \ne 0 \ne \map g z$.
Then:
:$\dfrac {\map {\paren {f g}' } z} {\map {\paren {f g} } z} = \dfrac{\map {f'} z} {\map f z} + \dfrac {\map {g'} z} {\map g z}$ | Follows directly from Product Rule for Complex Derivatives.
{{qed}} | Let $D \subset \C$ be [[Definition:Open Set (Complex Analysis)|open]].
Let $f, g: D \to \C$ be [[Definition:Analytic Function|analytic]].
Let $f g$ be their [[Definition:Pointwise Multiplication of Complex-Valued Functions|pointwise product]].
Let $z \in D$ with $\map f z \ne 0 \ne \map g z$.
Then:
:$\dfrac {\map ... | Follows directly from [[Product Rule for Complex Derivatives]].
{{qed}} | Logarithmic Derivative of Product of Analytic Functions | https://proofwiki.org/wiki/Logarithmic_Derivative_of_Product_of_Analytic_Functions | https://proofwiki.org/wiki/Logarithmic_Derivative_of_Product_of_Analytic_Functions | [
"Complex Analysis",
"Logarithmic Differentiation"
] | [
"Definition:Open Set/Complex Analysis",
"Definition:Analytic Function",
"Definition:Pointwise Multiplication of Complex-Valued Functions"
] | [
"Combination Theorem for Complex Derivatives/Product Rule"
] |
proofwiki-12983 | Uniformly Convergent Sequence on Dense Subset | Let $X$ be a metric space.
Let $Y \subset X$ be dense.
Let $V$ be a Banach space.
Let $\sequence {f_n}$ be a sequence of continuous mappings $f_n : X\to V$.
Let $\sequence {f_n}$ be uniformly convergent on $Y$.
Then $\sequence {f_n}$ is uniformly convergent on $X$. | Let $\epsilon>0$.
Let $N \in\N$ be such that $\norm {f_n - f_m} < \epsilon$ for $n, m > N$ on $Y$.
Let $x \in X$.
Then there exists a sequence $\sequence {y_n} \in Y$ with $y_n \to x$.
By continuity:
:$\norm {\map {f_n} x - \map {f_m} x} \le \epsilon$
Because $V$ is complete, $\sequence {f_n}$ is uniformly convergent o... | Let $X$ be a [[Definition:Metric Space|metric space]].
Let $Y \subset X$ be [[Definition:Dense Subset|dense]].
Let $V$ be a [[Definition:Banach Space|Banach space]].
Let $\sequence {f_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Continuous Mapping|continuous mappings]] $f_n : X\to V$.
Let $\sequence {f... | Let $\epsilon>0$.
Let $N \in\N$ be such that $\norm {f_n - f_m} < \epsilon$ for $n, m > N$ on $Y$.
Let $x \in X$.
Then there exists a [[Definition:Sequence|sequence]] $\sequence {y_n} \in Y$ with $y_n \to x$.
By continuity:
:$\norm {\map {f_n} x - \map {f_m} x} \le \epsilon$
Because $V$ is [[Definition:Complete Me... | Uniformly Convergent Sequence on Dense Subset | https://proofwiki.org/wiki/Uniformly_Convergent_Sequence_on_Dense_Subset | https://proofwiki.org/wiki/Uniformly_Convergent_Sequence_on_Dense_Subset | [
"Uniform Convergence"
] | [
"Definition:Metric Space",
"Definition:Everywhere Dense",
"Definition:Banach Space",
"Definition:Sequence",
"Definition:Continuous Mapping",
"Definition:Uniform Convergence",
"Definition:Uniform Convergence"
] | [
"Definition:Sequence",
"Definition:Complete Metric Space",
"Definition:Uniform Convergence",
"Category:Uniform Convergence"
] |
proofwiki-12984 | Factors in Uniformly Convergent Product Converge Uniformly to One | Let $X$ be a set.
Let $\struct {\mathbb K, \norm {\, \cdot \,} }$ be a valued field.
Let $\sequence {f_n}$ be a sequence of bounded mappings $f_n: X \to \mathbb K$.
Let the infinite product $\ds \prod_{n \mathop = 1}^\infty f_n$ converge uniformly on $X$.
Then $f_n$ converges uniformly to $1$. | Follows directly from Uniformly Convergent Product Satisfies Uniform Cauchy Criterion.
{{qed}} | Let $X$ be a [[Definition:Set|set]].
Let $\struct {\mathbb K, \norm {\, \cdot \,} }$ be a [[Definition:Valued Field|valued field]].
Let $\sequence {f_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Bounded Mapping|bounded mappings]] $f_n: X \to \mathbb K$.
Let the [[Definition:Infinite Product|infinite pro... | Follows directly from [[Uniformly Convergent Product Satisfies Uniform Cauchy Criterion]].
{{qed}} | Factors in Uniformly Convergent Product Converge Uniformly to One | https://proofwiki.org/wiki/Factors_in_Uniformly_Convergent_Product_Converge_Uniformly_to_One | https://proofwiki.org/wiki/Factors_in_Uniformly_Convergent_Product_Converge_Uniformly_to_One | [
"Uniform Convergence",
"Infinite Products"
] | [
"Definition:Set",
"Definition:Valued Field",
"Definition:Sequence",
"Definition:Bounded Mapping",
"Definition:Continued Product/Infinite",
"Definition:Uniform Convergence of Product",
"Definition:Uniform Convergence"
] | [
"Uniformly Convergent Product Satisfies Uniform Cauchy Criterion"
] |
proofwiki-12985 | Quintuplets of Consecutive Integers which are not Divisor Sum Values | The elements of the following $5$-tuples of consecutive integers have the property that they are not values of the divisor sum function $\map {\sigma_1} n$ for any $n$:
:$\tuple {49, 50, 51, 52, 53}$
:$\tuple {115, 116, 117, 118, 119}$
:$\tuple {145, 146, 147, 148, 149}$ | {{ProofWanted|Can be done by listing all $\sigma_1$s less than the numbers concerned, which is tedious}} | The [[Definition:Element|elements]] of the following [[Definition:Ordered Tuple|$5$-tuples]] of consecutive [[Definition:Positive Integer|integers]] have the property that they are not [[Definition:Value|values]] of the [[Definition:Divisor Sum Function|divisor sum function]] $\map {\sigma_1} n$ for any $n$:
:$\tuple ... | {{ProofWanted|Can be done by listing all $\sigma_1$s less than the numbers concerned, which is tedious}} | Quintuplets of Consecutive Integers which are not Divisor Sum Values | https://proofwiki.org/wiki/Quintuplets_of_Consecutive_Integers_which_are_not_Divisor_Sum_Values | https://proofwiki.org/wiki/Quintuplets_of_Consecutive_Integers_which_are_not_Divisor_Sum_Values | [
"Divisor Sum Function"
] | [
"Definition:Element",
"Definition:Ordered Tuple",
"Definition:Positive/Integer",
"Definition:Value",
"Definition:Divisor Sum Function"
] | [] |
proofwiki-12986 | Positive Integers which are not Divisor Sum Values | The following positive integers are not the values of the divisor sum function $\map {\sigma_1} n$ for any $n$:
:$2, 5, 9, 10, 11, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 33, 34, 35, 37, 41, 43, 45, 46, 47, 49, 50, 51, 52, 53, 55, 58, 59, 61, 64, 65, \ldots$
{{OEIS|A007369}} | {{ProofWanted|By exhaustion}}
Category:Divisor Sum Function
7od12pz6xwzbv54kx4p04gnwv37ng0u | The following [[Definition:Positive Integer|positive integers]] are not the [[Definition:Value|values]] of the [[Definition:Divisor Sum Function|divisor sum function]] $\map {\sigma_1} n$ for any $n$:
:$2, 5, 9, 10, 11, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 33, 34, 35, 37, 41, 43, 45, 46, 47, 49, 50, 51, 52, 53, 55, ... | {{ProofWanted|By exhaustion}}
[[Category:Divisor Sum Function]]
7od12pz6xwzbv54kx4p04gnwv37ng0u | Positive Integers which are not Divisor Sum Values | https://proofwiki.org/wiki/Positive_Integers_which_are_not_Divisor_Sum_Values | https://proofwiki.org/wiki/Positive_Integers_which_are_not_Divisor_Sum_Values | [
"Divisor Sum Function"
] | [
"Definition:Positive/Integer",
"Definition:Value",
"Definition:Divisor Sum Function"
] | [
"Category:Divisor Sum Function"
] |
proofwiki-12987 | Uniformly Absolutely Convergent Product is Uniformly Convergent | Let $X$ be a set.
Let $\struct {\mathbb K, \norm {\, \cdot \,} }$ be a valued field.
Let $\mathbb K$ be complete.
Let $\sequence {f_n}$ be a sequence of bounded mappings $f_n: X \to \mathbb K$.
Let the infinite product $\ds \prod_{n \mathop = 1}^\infty f_n$ converge uniformly absolutely on $X$.
Then it converges unifor... | {{ProofWanted}}
Category:Uniform Convergence
Category:Infinite Products
enywz0kpgqttb7mm01wfxavpe4axpw7 | Let $X$ be a [[Definition:Set|set]].
Let $\struct {\mathbb K, \norm {\, \cdot \,} }$ be a [[Definition:Valued Field|valued field]].
Let $\mathbb K$ be [[Definition:Complete Metric Space|complete]].
Let $\sequence {f_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Bounded Mapping|bounded mappings]] $f_n: X ... | {{ProofWanted}}
[[Category:Uniform Convergence]]
[[Category:Infinite Products]]
enywz0kpgqttb7mm01wfxavpe4axpw7 | Uniformly Absolutely Convergent Product is Uniformly Convergent | https://proofwiki.org/wiki/Uniformly_Absolutely_Convergent_Product_is_Uniformly_Convergent | https://proofwiki.org/wiki/Uniformly_Absolutely_Convergent_Product_is_Uniformly_Convergent | [
"Uniform Convergence",
"Infinite Products"
] | [
"Definition:Set",
"Definition:Valued Field",
"Definition:Complete Metric Space",
"Definition:Sequence",
"Definition:Bounded Mapping",
"Definition:Continued Product/Infinite",
"Definition:Uniform Absolute Convergence of Product",
"Definition:Uniform Convergence of Product"
] | [
"Category:Uniform Convergence",
"Category:Infinite Products"
] |
proofwiki-12988 | Zeroes of Infinite Product of Analytic Functions | Let $D \subset \C$ be an open connected set.
Let $\sequence {f_n}$ be a sequence of analytic functions $f_n: D \to \C$.
Let $\ds \prod_{n \mathop = 1}^\infty f_n$ converge locally uniformly to $f$.
Let $z_0\in D$.
Then:
:$(1): \quad$ $f$ is identically zero {{iff}} some $f_n$ is identically zero
:$(2): \quad$ $\map {f... | Note that by Infinite Product of Analytic Functions is Analytic, $f$ is analytic.
Let $n_0 \in \N$ and $U\subset D$ an open neighborhood of $z_0$ such that $\ds \prod_{n \mathop = n_0}^\infty \map {f_n} z \ne 0$ for $z\in U$.
Let $f$ be identically zero on $U$.
Then $\ds \prod_{n \mathop = 1}^{n_0 - 1} \map {f_n} z$ is... | Let $D \subset \C$ be an [[Definition:Open Set (Complex Analysis)|open]] [[Definition:Connected Topological Space|connected set]].
Let $\sequence {f_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Analytic Function|analytic functions]] $f_n: D \to \C$.
Let $\ds \prod_{n \mathop = 1}^\infty f_n$ [[Definition... | Note that by [[Infinite Product of Analytic Functions is Analytic]], $f$ is [[Definition:Analytic Function|analytic]].
Let $n_0 \in \N$ and $U\subset D$ an [[Definition:Open Neighborhood (Complex Analysis)|open neighborhood]] of $z_0$ such that $\ds \prod_{n \mathop = n_0}^\infty \map {f_n} z \ne 0$ for $z\in U$.
Let... | Zeroes of Infinite Product of Analytic Functions | https://proofwiki.org/wiki/Zeroes_of_Infinite_Product_of_Analytic_Functions | https://proofwiki.org/wiki/Zeroes_of_Infinite_Product_of_Analytic_Functions | [
"Infinite Products"
] | [
"Definition:Open Set/Complex Analysis",
"Definition:Connected Topological Space",
"Definition:Sequence",
"Definition:Analytic Function",
"Definition:Locally Uniform Convergence of Product",
"Definition:Identically Zero",
"Definition:Identically Zero",
"Definition:Finite Set",
"Definition:Multiplicit... | [
"Infinite Product of Analytic Functions is Analytic",
"Definition:Analytic Function",
"Definition:Open Neighborhood (Complex Analysis)",
"Definition:Identically Zero",
"Definition:Identically Zero",
"Definition:Identically Zero",
"Uniqueness of Analytic Continuation",
"Definition:Identically Zero",
... |
proofwiki-12989 | Infinite Product of Analytic Functions is Analytic | Let $D \subset \C$ be an open set.
Let $\sequence {f_n}$ be a sequence of analytic functions $f_n: D \to \C$.
Let $\ds \prod_{n \mathop = 1}^\infty f_n$ converge locally uniformly to $f$.
Then $f$ is analytic. | Follows directly from:
:Partial Products of Uniformly Convergent Product Converge Uniformly
:Uniform Limit of Analytic Functions is Analytic
{{qed}} | Let $D \subset \C$ be an [[Definition:Open Set (Complex Analysis)|open set]].
Let $\sequence {f_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Analytic Function|analytic functions]] $f_n: D \to \C$.
Let $\ds \prod_{n \mathop = 1}^\infty f_n$ [[Definition:Locally Uniform Convergence of Product|converge loca... | Follows directly from:
:[[Partial Products of Uniformly Convergent Product Converge Uniformly]]
:[[Uniform Limit of Analytic Functions is Analytic]]
{{qed}} | Infinite Product of Analytic Functions is Analytic | https://proofwiki.org/wiki/Infinite_Product_of_Analytic_Functions_is_Analytic | https://proofwiki.org/wiki/Infinite_Product_of_Analytic_Functions_is_Analytic | [
"Infinite Products"
] | [
"Definition:Open Set/Complex Analysis",
"Definition:Sequence",
"Definition:Analytic Function",
"Definition:Locally Uniform Convergence of Product",
"Definition:Analytic Function"
] | [
"Partial Products of Uniformly Convergent Product Converge Uniformly",
"Uniform Limit of Analytic Functions is Analytic"
] |
proofwiki-12990 | Sum of 2 Squares in 2 Distinct Ways | Let $m, n \in \Z_{>0}$ be distinct positive integers that can be expressed as the sum of two distinct square numbers.
Then $m n$ can be expressed as the sum of two square numbers in at least two distinct ways. | Let:
:$m = a^2 + b^2$
:$n = c^2 + d^2$
Then:
{{begin-eqn}}
{{eqn | l = m n
| r = \paren {a^2 + b^2} \paren {c^2 + d^2}
| c =
}}
{{eqn | r = \paren {a c + b d}^2 + \paren {a d - b c}^2
| c = Brahmagupta-Fibonacci Identity
}}
{{eqn | r = \paren {a c - b d}^2 + \paren {a d + b c}^2
| c = {{Corolla... | Let $m, n \in \Z_{>0}$ be [[Definition:Distinct|distinct]] [[Definition:Positive Integer|positive integers]] that can be expressed as the [[Definition:Integer Addition|sum]] of two [[Definition:Distinct|distinct]] [[Definition:Square Number|square numbers]].
Then $m n$ can be expressed as the [[Definition:Integer Addi... | Let:
:$m = a^2 + b^2$
:$n = c^2 + d^2$
Then:
{{begin-eqn}}
{{eqn | l = m n
| r = \paren {a^2 + b^2} \paren {c^2 + d^2}
| c =
}}
{{eqn | r = \paren {a c + b d}^2 + \paren {a d - b c}^2
| c = [[Brahmagupta-Fibonacci Identity]]
}}
{{eqn | r = \paren {a c - b d}^2 + \paren {a d + b c}^2
| c = {{C... | Sum of 2 Squares in 2 Distinct Ways | https://proofwiki.org/wiki/Sum_of_2_Squares_in_2_Distinct_Ways | https://proofwiki.org/wiki/Sum_of_2_Squares_in_2_Distinct_Ways | [
"Sum of 2 Squares in 2 Distinct Ways",
"Sums of Squares",
"Square Numbers",
"Brahmagupta-Fibonacci Identity"
] | [
"Definition:Distinct",
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Distinct",
"Definition:Square Number",
"Definition:Addition/Integers",
"Definition:Square Number",
"Definition:Distinct"
] | [
"Brahmagupta-Fibonacci Identity"
] |
proofwiki-12991 | Bounds for Finite Product of Real Numbers | Let $a_1, a_2, \ldots, a_n$ be positive real numbers.
Then:
:$\ds \sum_{k \mathop = 1}^n a_k \le \prod_{k \mathop = 1}^n \paren {1 + a_k} \le \map \exp {\sum_{k \mathop = 1}^n a_k}$ | === Lower bound ===
Follows by expanding.
{{qed|lemma}} | Let $a_1, a_2, \ldots, a_n$ be [[Definition:Positive Real Number|positive real numbers]].
Then:
:$\ds \sum_{k \mathop = 1}^n a_k \le \prod_{k \mathop = 1}^n \paren {1 + a_k} \le \map \exp {\sum_{k \mathop = 1}^n a_k}$ | === Lower bound ===
Follows by expanding.
{{qed|lemma}} | Bounds for Finite Product of Real Numbers | https://proofwiki.org/wiki/Bounds_for_Finite_Product_of_Real_Numbers | https://proofwiki.org/wiki/Bounds_for_Finite_Product_of_Real_Numbers | [
"Inequalities"
] | [
"Definition:Positive/Real Number"
] | [] |
proofwiki-12992 | Bounds for Complex Exponential | Let $\exp$ denote the complex exponential.
Let $z \in \C$ with $\cmod z \le \dfrac 1 2$.
Then
:$\dfrac 1 2 \cmod z \le \cmod {\exp z - 1} \le \dfrac 3 2 \cmod z$ | By definition of complex exponential:
:$\exp z = \ds \sum_{n \mathop = 1}^\infty \frac {z^n} {n!}$
Thus
{{begin-eqn}}
{{eqn | l = \cmod {\exp z - 1 - z}
| r = \cmod {\sum_{n \mathop = 2}^\infty \frac {z^n} {n!} }
| c = Linear Combination of Convergent Series
}}
{{eqn | o = \le
| r = \sum_{n \math... | Let $\exp$ denote the [[Definition:Complex Exponential Function|complex exponential]].
Let $z \in \C$ with $\cmod z \le \dfrac 1 2$.
Then
:$\dfrac 1 2 \cmod z \le \cmod {\exp z - 1} \le \dfrac 3 2 \cmod z$ | By definition of [[Definition:Complex Exponential Function|complex exponential]]:
:$\exp z = \ds \sum_{n \mathop = 1}^\infty \frac {z^n} {n!}$
Thus
{{begin-eqn}}
{{eqn | l = \cmod {\exp z - 1 - z}
| r = \cmod {\sum_{n \mathop = 2}^\infty \frac {z^n} {n!} }
| c = [[Linear Combination of Convergent Serie... | Bounds for Complex Exponential | https://proofwiki.org/wiki/Bounds_for_Complex_Exponential | https://proofwiki.org/wiki/Bounds_for_Complex_Exponential | [
"Exponential Function"
] | [
"Definition:Exponential Function/Complex"
] | [
"Definition:Exponential Function/Complex",
"Linear Combination of Convergent Series",
"Triangle Inequality for Series",
"Sum of Infinite Geometric Sequence",
"Triangle Inequality"
] |
proofwiki-12993 | Sum of 2 Squares in 2 Distinct Ways/Examples/50 | $50$ is the smallest positive integer which can be expressed as the sum of two square numbers in two distinct ways:
{{begin-eqn}}
{{eqn | l = 50
| r = 5^2 + 5^2
}}
{{eqn | r = 7^2 + 1^2
}}
{{end-eqn}} | The smallest two positive integers which can be expressed as the sum of two distinct square numbers are:
{{begin-eqn}}
{{eqn | l = 5
| r = 1^2 + 2^2
}}
{{eqn | l = 10
| r = 1^2 + 3^2
}}
{{end-eqn}}
We have that:
:$50 = 5 \times 10$
Thus:
{{begin-eqn}}
{{eqn | r = \paren {1^2 + 2^2} \paren {1^2 + 3^2}
... | $50$ is the smallest [[Definition:Positive Integer|positive integer]] which can be expressed as the [[Definition:Integer Addition|sum]] of two [[Definition:Square Number|square numbers]] in two [[Definition:Distinct|distinct]] ways:
{{begin-eqn}}
{{eqn | l = 50
| r = 5^2 + 5^2
}}
{{eqn | r = 7^2 + 1^2
}}
{{end-e... | The smallest two [[Definition:Positive Integer|positive integers]] which can be expressed as the [[Definition:Integer Addition|sum]] of two [[Definition:Distinct|distinct]] [[Definition:Square Number|square numbers]] are:
{{begin-eqn}}
{{eqn | l = 5
| r = 1^2 + 2^2
}}
{{eqn | l = 10
| r = 1^2 + 3^2
}}
{{en... | Sum of 2 Squares in 2 Distinct Ways/Examples/50 | https://proofwiki.org/wiki/Sum_of_2_Squares_in_2_Distinct_Ways/Examples/50 | https://proofwiki.org/wiki/Sum_of_2_Squares_in_2_Distinct_Ways/Examples/50 | [
"Sum of 2 Squares in 2 Distinct Ways",
"50"
] | [
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Square Number",
"Definition:Distinct"
] | [
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Distinct",
"Definition:Square Number",
"Brahmagupta-Fibonacci Identity"
] |
proofwiki-12994 | Sum of 2 Squares in 2 Distinct Ways/Examples/65 | $65$ can be expressed as the sum of two square numbers in two distinct ways:
{{begin-eqn}}
{{eqn | l = 65
| r = 8^2 + 1^2
}}
{{eqn | r = 7^2 + 4^2
}}
{{end-eqn}} | We have that:
:$65 = 5 \times 13$
Both $5$ and $13$ can be expressed as the sum of two distinct square numbers:
{{begin-eqn}}
{{eqn | l = 5
| r = 1^2 + 2^2
}}
{{eqn | l = 13
| r = 2^2 + 3^2
}}
{{end-eqn}}
Thus:
{{begin-eqn}}
{{eqn | r = \paren {1^2 + 2^2} \paren {2^2 + 3^2}
| c =
}}
{{eqn | r = \pare... | $65$ can be expressed as the [[Definition:Integer Addition|sum]] of two [[Definition:Square Number|square numbers]] in two [[Definition:Distinct|distinct]] ways:
{{begin-eqn}}
{{eqn | l = 65
| r = 8^2 + 1^2
}}
{{eqn | r = 7^2 + 4^2
}}
{{end-eqn}} | We have that:
:$65 = 5 \times 13$
Both $5$ and $13$ can be expressed as the [[Definition:Integer Addition|sum]] of two [[Definition:Distinct|distinct]] [[Definition:Square Number|square numbers]]:
{{begin-eqn}}
{{eqn | l = 5
| r = 1^2 + 2^2
}}
{{eqn | l = 13
| r = 2^2 + 3^2
}}
{{end-eqn}}
Thus:
{{begin... | Sum of 2 Squares in 2 Distinct Ways/Examples/65 | https://proofwiki.org/wiki/Sum_of_2_Squares_in_2_Distinct_Ways/Examples/65 | https://proofwiki.org/wiki/Sum_of_2_Squares_in_2_Distinct_Ways/Examples/65 | [
"Sum of 2 Squares in 2 Distinct Ways",
"65"
] | [
"Definition:Addition/Integers",
"Definition:Square Number",
"Definition:Distinct"
] | [
"Definition:Addition/Integers",
"Definition:Distinct",
"Definition:Square Number",
"Brahmagupta-Fibonacci Identity"
] |
proofwiki-12995 | Continuous Implies Locally Bounded | Let $X$ be a topological space.
Let $M$ be a metric space.
Let $f: X \to M$ be continuous.
Then $f$ is locally bounded. | Let $x \in X$.
Let $U = \map {f^{-1} } {\map B {\map f x, 1} }$.
{{explain|$\map B {\map f x, 1}$}}
By continuity, $U$ is a neighborhood of $x$.
Because $\map f U \subset \map B {\map f x, 1}$, $f$ is bounded on $U$.
Thus $f$ is locally bounded.
{{qed}}
Category:Boundedness
032x9q3dgsc1qdh2h7vr31bk99m0l6f | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $M$ be a [[Definition:Metric Space|metric space]].
Let $f: X \to M$ be [[Definition:Continuous Mapping|continuous]].
Then $f$ is [[Definition:Locally Bounded Mapping|locally bounded]]. | Let $x \in X$.
Let $U = \map {f^{-1} } {\map B {\map f x, 1} }$.
{{explain|$\map B {\map f x, 1}$}}
By [[Definition:Continuous Mapping|continuity]], $U$ is a [[Definition:Neighborhood of Point in Topological Space|neighborhood]] of $x$.
Because $\map f U \subset \map B {\map f x, 1}$, $f$ is [[Definition:Bounded Ma... | Continuous Implies Locally Bounded | https://proofwiki.org/wiki/Continuous_Implies_Locally_Bounded | https://proofwiki.org/wiki/Continuous_Implies_Locally_Bounded | [
"Boundedness"
] | [
"Definition:Topological Space",
"Definition:Metric Space",
"Definition:Continuous Mapping",
"Definition:Locally Bounded/Mapping"
] | [
"Definition:Continuous Mapping",
"Definition:Neighborhood (Topology)/Point",
"Definition:Bounded Mapping",
"Definition:Locally Bounded/Mapping",
"Category:Boundedness"
] |
proofwiki-12996 | Lower Adjoint at Element is Minimum of Preimage of Singleton of Element implies Composition is Identity | Let $L = \struct {S, \preceq}, R = \struct {T, \precsim}$ be ordered sets.
Let $g: S \to T, d: T \to S$ be mappings such that
:$\forall t \in T: \map d t = \min \set {g^{-1} \sqbrk {\set t} }$
Then $g \circ d = I_T$
where $I_T$ denotes the identity mapping of $T$. | Let $t \in T$.
By assumption:
:$\map d t = \min \set {g^{-1} \sqbrk {\set t} }$
By definition of min operation:
:$\map d t = \map \inf {g^{-1} \sqbrk {\set t} }$ and $\map d t \in g^{-1} \sqbrk {\set t}$
By definition of image of set:
:$\map g {\map d t} \in \set t$
By definition of singleton:
:$\map g {\map d t} = t$
... | Let $L = \struct {S, \preceq}, R = \struct {T, \precsim}$ be [[Definition:Ordered Set|ordered sets]].
Let $g: S \to T, d: T \to S$ be [[Definition:Mapping|mappings]] such that
:$\forall t \in T: \map d t = \min \set {g^{-1} \sqbrk {\set t} }$
Then $g \circ d = I_T$
where $I_T$ denotes the [[Definition:Identity Mapp... | Let $t \in T$.
By assumption:
:$\map d t = \min \set {g^{-1} \sqbrk {\set t} }$
By definition of [[Definition:Min Operation|min operation]]:
:$\map d t = \map \inf {g^{-1} \sqbrk {\set t} }$ and $\map d t \in g^{-1} \sqbrk {\set t}$
By definition of [[Definition:Image of Subset under Mapping|image of set]]:
:$\map g... | Lower Adjoint at Element is Minimum of Preimage of Singleton of Element implies Composition is Identity | https://proofwiki.org/wiki/Lower_Adjoint_at_Element_is_Minimum_of_Preimage_of_Singleton_of_Element_implies_Composition_is_Identity | https://proofwiki.org/wiki/Lower_Adjoint_at_Element_is_Minimum_of_Preimage_of_Singleton_of_Element_implies_Composition_is_Identity | [
"Galois Connections"
] | [
"Definition:Ordered Set",
"Definition:Mapping",
"Definition:Identity Mapping"
] | [
"Definition:Min Operation",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Singleton",
"Definition:Composition of Mappings"
] |
proofwiki-12997 | Bounds for Weierstrass Elementary Factors | Let $E_p: \C \to \C$ denote the $p$th Weierstrass elementary factor:
$\quad\map {E_p} z = \begin {cases} 1 - z & : p = 0 \\
\paren {1 - z} \map \exp {z + \dfrac {z^2} 2 + \cdots + \dfrac {z^p} p} & : \text {otherwise} \end {cases}$
Let $z \in \C$. | === Proof of some bound ===
Let $\cmod z \le \dfrac 1 2$.
We may assume $p \ge 1$.
We have:
:$\map {E_p} z = \map \exp {\map \log {1 - z} + \ds \sum_{k \mathop = 1}^p \frac {z^k} k}$
Then:
{{begin-eqn}}
{{eqn | l = \cmod {\map \log {1 - z} + \sum_{k \mathop = 1}^p \frac {z^k} k}
| r = \cmod {-\sum_{k \mathop = p ... | Let $E_p: \C \to \C$ denote the $p$th [[Definition:Weierstrass Elementary Factor|Weierstrass elementary factor]]:
$\quad\map {E_p} z = \begin {cases} 1 - z & : p = 0 \\
\paren {1 - z} \map \exp {z + \dfrac {z^2} 2 + \cdots + \dfrac {z^p} p} & : \text {otherwise} \end {cases}$
Let $z \in \C$. | === Proof of some bound ===
Let $\cmod z \le \dfrac 1 2$.
We may assume $p \ge 1$.
We have:
:$\map {E_p} z = \map \exp {\map \log {1 - z} + \ds \sum_{k \mathop = 1}^p \frac {z^k} k}$
Then:
{{begin-eqn}}
{{eqn | l = \cmod {\map \log {1 - z} + \sum_{k \mathop = 1}^p \frac {z^k} k}
| r = \cmod {-\sum_{k \mathop... | Bounds for Weierstrass Elementary Factors | https://proofwiki.org/wiki/Bounds_for_Weierstrass_Elementary_Factors | https://proofwiki.org/wiki/Bounds_for_Weierstrass_Elementary_Factors | [
"Infinite Products"
] | [
"Definition:Weierstrass Elementary Factor"
] | [
"Series Expansion of Complex Logarithm",
"Triangle Inequality for Series",
"Sum of Infinite Geometric Sequence",
"Bounds for Complex Exponential",
"Triangle Inequality for Series"
] |
proofwiki-12998 | Order of Product of Entire Functions | Let $f, g: \C \to \C$ be entire functions of order $\alpha$ and $\beta$.
Then $f g$ has order at most $\map \max {\alpha, \beta}$. | If $\map \max {\alpha, \beta} = +\infty$, the claim is trivial.
Thus we may assume that $\alpha < +\infty$ and $\beta < +\infty$.
Let $\epsilon > 0$ be arbitrary.
By {{Defof|Order of Entire Function|index=1}}, we have:
:$\map f z = \map \OO {\map \exp {\cmod z^{\alpha + \epsilon} } }$
and:
:$\map g z = \map \OO {\map \... | Let $f, g: \C \to \C$ be [[Definition:Entire Function|entire functions]] of [[Definition:Order of Entire Function|order]] $\alpha$ and $\beta$.
Then $f g$ has [[Definition:Order of Entire Function|order]] at most $\map \max {\alpha, \beta}$. | If $\map \max {\alpha, \beta} = +\infty$, the claim is trivial.
Thus we may assume that $\alpha < +\infty$ and $\beta < +\infty$.
Let $\epsilon > 0$ be arbitrary.
By {{Defof|Order of Entire Function|index=1}}, we have:
:$\map f z = \map \OO {\map \exp {\cmod z^{\alpha + \epsilon} } }$
and:
:$\map g z = \map \OO {\m... | Order of Product of Entire Functions | https://proofwiki.org/wiki/Order_of_Product_of_Entire_Functions | https://proofwiki.org/wiki/Order_of_Product_of_Entire_Functions | [
"Entire Functions"
] | [
"Definition:Entire Function",
"Definition:Order of Entire Function",
"Definition:Order of Entire Function"
] | [
"Definition:Order of Entire Function"
] |
proofwiki-12999 | Numbers Partitioned into up to 4 Squares in 5 Ways | The following positive integers can be expressed as the sum of no more than $4$ squares in $5$ distinct ways:
:$50, 52, 54, 58, \ldots$ | {{begin-eqn}}
{{eqn | l = 50
| r = 7^2 + 1^2
| c =
}}
{{eqn | r = 5^2 + 5^2
| c =
}}
{{eqn | r = 5^2 + 4^2 + 3^2
| c =
}}
{{eqn | r = 4^2 + 4^2 + 3^2 + 3^2
| c =
}}
{{eqn | r = 6^2 + 3^2 + 2^2 + 1^2
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 52
| r = 7^2 + 1^2 + 1^2 + ... | The following [[Definition:Positive Integer|positive integers]] can be expressed as the [[Definition:Integer Addition|sum]] of no more than $4$ [[Definition:Square Number|squares]] in $5$ [[Definition:Distinct|distinct]] ways:
:$50, 52, 54, 58, \ldots$ | {{begin-eqn}}
{{eqn | l = 50
| r = 7^2 + 1^2
| c =
}}
{{eqn | r = 5^2 + 5^2
| c =
}}
{{eqn | r = 5^2 + 4^2 + 3^2
| c =
}}
{{eqn | r = 4^2 + 4^2 + 3^2 + 3^2
| c =
}}
{{eqn | r = 6^2 + 3^2 + 2^2 + 1^2
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 52
| r = 7^2 + 1^2 + 1^2 ... | Numbers Partitioned into up to 4 Squares in 5 Ways | https://proofwiki.org/wiki/Numbers_Partitioned_into_up_to_4_Squares_in_5_Ways | https://proofwiki.org/wiki/Numbers_Partitioned_into_up_to_4_Squares_in_5_Ways | [
"Sums of Squares"
] | [
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Square Number",
"Definition:Distinct"
] | [] |
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