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-17800 | Partial Derivative wrt x of sin x y over cos (x + y) | :$\dfrac \partial {\partial x} \dfrac {\sin x y} {\map \cos {x + y} } = \dfrac {y \map \cos {x + y} \cos x y + \map \sin {x + y} \sin x y} {\map {\cos^2} {x + y} }$ | {{begin-eqn}}
{{eqn | l = \frac \partial {\partial x} \frac {\sin x y} {\map \cos {x + y} }
| r = \frac {\map \cos {x + y} y \cos x y - \sin x y \paren {-\map \sin {x + y} } } {\map {\cos^2} {x + y} }
| c = Quotient Rule for Derivatives, treating $y$ as a constant
}}
{{eqn | r = \dfrac {y \map \cos {x + y} ... | :$\dfrac \partial {\partial x} \dfrac {\sin x y} {\map \cos {x + y} } = \dfrac {y \map \cos {x + y} \cos x y + \map \sin {x + y} \sin x y} {\map {\cos^2} {x + y} }$ | {{begin-eqn}}
{{eqn | l = \frac \partial {\partial x} \frac {\sin x y} {\map \cos {x + y} }
| r = \frac {\map \cos {x + y} y \cos x y - \sin x y \paren {-\map \sin {x + y} } } {\map {\cos^2} {x + y} }
| c = [[Quotient Rule for Derivatives]], treating $y$ as a [[Definition:Constant|constant]]
}}
{{eqn | r = ... | Partial Derivative wrt x of sin x y over cos (x + y) | https://proofwiki.org/wiki/Partial_Derivative_wrt_x_of_sin_x_y_over_cos_(x_+_y) | https://proofwiki.org/wiki/Partial_Derivative_wrt_x_of_sin_x_y_over_cos_(x_+_y) | [
"Examples of Partial Derivatives"
] | [] | [
"Quotient Rule for Derivatives",
"Definition:Constant"
] |
proofwiki-17801 | Second Partial Derivative wrt r of ln (r^2 + s) | :$\dfrac {\partial^2} {\partial r^2} \map \ln {r^2 + s} = \dfrac {2 \paren {s - r^2} } {\paren {r^2 + s}^2}$ | {{begin-eqn}}
{{eqn | l = \dfrac {\partial^2} {\partial r^2} \map \ln {r^2 + s}
| r = \map {\dfrac \partial {\partial r} } {\dfrac \partial {\partial r} \map \ln {r^2 + s} }
| c = {{Defof|Second Partial Derivative}}
}}
{{eqn | r = \dfrac \partial {\partial r} \frac 1 {r^2 + s} 2 r
| c = Derivative of ... | :$\dfrac {\partial^2} {\partial r^2} \map \ln {r^2 + s} = \dfrac {2 \paren {s - r^2} } {\paren {r^2 + s}^2}$ | {{begin-eqn}}
{{eqn | l = \dfrac {\partial^2} {\partial r^2} \map \ln {r^2 + s}
| r = \map {\dfrac \partial {\partial r} } {\dfrac \partial {\partial r} \map \ln {r^2 + s} }
| c = {{Defof|Second Partial Derivative}}
}}
{{eqn | r = \dfrac \partial {\partial r} \frac 1 {r^2 + s} 2 r
| c = [[Derivative o... | Second Partial Derivative wrt r of ln (r^2 + s) | https://proofwiki.org/wiki/Second_Partial_Derivative_wrt_r_of_ln_(r^2_+_s) | https://proofwiki.org/wiki/Second_Partial_Derivative_wrt_r_of_ln_(r^2_+_s) | [
"Examples of Partial Derivatives"
] | [] | [
"Derivative of Natural Logarithm Function",
"Derivative of Composite Function",
"Definition:Constant",
"Quotient Rule for Derivatives",
"Definition:Constant"
] |
proofwiki-17802 | Partial Derivatives of tan^2 (x^2 - y^2) | Let:
:$\map f {x, y} = \map {\tan^2} {x^2 - y^2}$
Then:
{{begin-eqn}}
{{eqn | l = \map {f_1} {x, y}
| r = 4 x \map \tan {x^2 - y^2} \map {\sec^2} {x^2 - y^2}
}}
{{eqn | l = \map {f_2} {1, 2}
| r = 8 \tan 3 \sec^2 3
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \map {f_1} {x, y}
| r = \dfrac \partial {\partial x} {\map {\tan^2} {x^2 - y^2} }
| c = {{Defof|Partial Derivative}}
}}
{{eqn | r = 2 \map \tan {x^2 - y^2} \map {\sec^2} {x^2 - y^2} \cdot 2 x
| c = Derivative of Square of Tangent, Derivative of Square Function, Chain Rule for... | Let:
:$\map f {x, y} = \map {\tan^2} {x^2 - y^2}$
Then:
{{begin-eqn}}
{{eqn | l = \map {f_1} {x, y}
| r = 4 x \map \tan {x^2 - y^2} \map {\sec^2} {x^2 - y^2}
}}
{{eqn | l = \map {f_2} {1, 2}
| r = 8 \tan 3 \sec^2 3
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \map {f_1} {x, y}
| r = \dfrac \partial {\partial x} {\map {\tan^2} {x^2 - y^2} }
| c = {{Defof|Partial Derivative}}
}}
{{eqn | r = 2 \map \tan {x^2 - y^2} \map {\sec^2} {x^2 - y^2} \cdot 2 x
| c = [[Derivative of Square of Tangent]], [[Derivative of Square Function]], [[Chai... | Partial Derivatives of tan^2 (x^2 - y^2) | https://proofwiki.org/wiki/Partial_Derivatives_of_tan^2_(x^2_-_y^2) | https://proofwiki.org/wiki/Partial_Derivatives_of_tan^2_(x^2_-_y^2) | [
"Examples of Partial Derivatives"
] | [] | [
"Derivative of Square of Tangent",
"Derivative of Square Function",
"Derivative of Composite Function",
"Derivative of Square of Tangent",
"Derivative of Square Function",
"Derivative of Composite Function",
"Tangent Function is Odd",
"Secant Function is Even"
] |
proofwiki-17803 | Derivative of Square of Tangent | :$\map {\dfrac \d {\d x} } {\tan^2 x} = 2 \tan x \sec^2 x$
when $\cos x \ne 0$. | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\tan^2 x}
| r = 2 \tan x \map {\dfrac \d {\d x} } {\tan x}
| c = Chain Rule for Derivatives, Derivative of Square Function
}}
{{eqn | r = 2 \tan x \sec^2 x
| c = Derivative of Tangent Function
}}
{{end-eqn}}
{{qed}}
Category:Tangent Function
nr7j1qx... | :$\map {\dfrac \d {\d x} } {\tan^2 x} = 2 \tan x \sec^2 x$
when $\cos x \ne 0$. | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\tan^2 x}
| r = 2 \tan x \map {\dfrac \d {\d x} } {\tan x}
| c = [[Chain Rule for Derivatives]], [[Derivative of Square Function]]
}}
{{eqn | r = 2 \tan x \sec^2 x
| c = [[Derivative of Tangent Function]]
}}
{{end-eqn}}
{{qed}}
[[Category:Tangent F... | Derivative of Square of Tangent | https://proofwiki.org/wiki/Derivative_of_Square_of_Tangent | https://proofwiki.org/wiki/Derivative_of_Square_of_Tangent | [
"Tangent Function"
] | [] | [
"Derivative of Composite Function",
"Derivative of Square Function",
"Derivative of Tangent Function",
"Category:Tangent Function"
] |
proofwiki-17804 | Partial Derivative wrt z of z^2 equals x^2 - 2 x y - 1 at (1, -2, -2) | Let $z^2 = x^2 - 2 x y - 1$.
Then:
:$\valueat {\dfrac {\partial z} {\partial x} } {x \mathop = 1, y \mathop = -2, z \mathop = -2} = -\dfrac 3 2$ | First we make sure that $\tuple {1, -2, -2}$ actually satisfies the equation:
{{begin-eqn}}
{{eqn | l = x^2 - 2 x y - 1
| r = \paren 1^2 - 2 \paren 1 \paren {-2} - 1
| c = at $\tuple {1, -2, -2}$
}}
{{eqn | r = 1 + 4 - 1
| c =
}}
{{eqn | r = 4
| c =
}}
{{eqn | r = \paren {-2}^2
| c =
}}... | Let $z^2 = x^2 - 2 x y - 1$.
Then:
:$\valueat {\dfrac {\partial z} {\partial x} } {x \mathop = 1, y \mathop = -2, z \mathop = -2} = -\dfrac 3 2$ | First we make sure that $\tuple {1, -2, -2}$ actually satisfies the equation:
{{begin-eqn}}
{{eqn | l = x^2 - 2 x y - 1
| r = \paren 1^2 - 2 \paren 1 \paren {-2} - 1
| c = at $\tuple {1, -2, -2}$
}}
{{eqn | r = 1 + 4 - 1
| c =
}}
{{eqn | r = 4
| c =
}}
{{eqn | r = \paren {-2}^2
| c =
}... | Partial Derivative wrt z of z^2 equals x^2 - 2 x y - 1 at (1, -2, -2) | https://proofwiki.org/wiki/Partial_Derivative_wrt_z_of_z^2_equals_x^2_-_2_x_y_-_1_at_(1,_-2,_-2) | https://proofwiki.org/wiki/Partial_Derivative_wrt_z_of_z^2_equals_x^2_-_2_x_y_-_1_at_(1,_-2,_-2) | [
"Partial Derivative wrt z of z^2 equals x^2 - 2 x y - 1 at (1, -2, -2)",
"Examples of Partial Derivatives"
] | [] | [] |
proofwiki-17805 | Partial Derivative/Examples/u - v + 2 w, 2 u + v + 2 w, u - v + w | Let:
{{begin-eqn}}
{{eqn | l = u - v + 2 w
| r = x + 2 z
}}
{{eqn | l = 2 u + v - 2 w
| r = 2 x - 2 z
}}
{{eqn | l = u - v + w
| r = z - y
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \dfrac {\partial u} {\partial y}
| r = 0
}}
{{eqn | l = \dfrac {\partial v} {\partial y}
| r = 2
}}
{{eq... | Partial differentiation {{WRT|Differentiation}} $y$ gives:
{{begin-eqn}}
{{eqn | l = \dfrac {\partial u} {\partial y} - \dfrac {\partial v} {\partial y} + 2 \dfrac {\partial w} {\partial y}
| r = 0
}}
{{eqn | l = 2 \dfrac {\partial u} {\partial y} + \dfrac {\partial v} {\partial y} - 2 \dfrac {\partial w} {\parti... | Let:
{{begin-eqn}}
{{eqn | l = u - v + 2 w
| r = x + 2 z
}}
{{eqn | l = 2 u + v - 2 w
| r = 2 x - 2 z
}}
{{eqn | l = u - v + w
| r = z - y
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \dfrac {\partial u} {\partial y}
| r = 0
}}
{{eqn | l = \dfrac {\partial v} {\partial y}
| r = 2
}}
{... | [[Definition:Partial Derivative|Partial differentiation]] {{WRT|Differentiation}} $y$ gives:
{{begin-eqn}}
{{eqn | l = \dfrac {\partial u} {\partial y} - \dfrac {\partial v} {\partial y} + 2 \dfrac {\partial w} {\partial y}
| r = 0
}}
{{eqn | l = 2 \dfrac {\partial u} {\partial y} + \dfrac {\partial v} {\partial... | Partial Derivative/Examples/u - v + 2 w, 2 u + v + 2 w, u - v + w | https://proofwiki.org/wiki/Partial_Derivative/Examples/u_-_v_+_2_w,_2_u_+_v_+_2_w,_u_-_v_+_w | https://proofwiki.org/wiki/Partial_Derivative/Examples/u_-_v_+_2_w,_2_u_+_v_+_2_w,_u_-_v_+_w | [
"Examples of Partial Derivatives"
] | [] | [
"Definition:Partial Derivative",
"Definition:Matrix",
"Cramer's Rule",
"Cramer's Rule"
] |
proofwiki-17806 | Partial Derivatives of x tan^-1 (x^2 + y) | Let:
:$\map f {x, y} = x \map \arctan {x^2 + y}$
Then:
{{begin-eqn}}
{{eqn | l = \map {f_1} {1, 0}
| r = \dfrac \pi 4 + 1
}}
{{eqn | l = \map {f_2} {x, y}
| r = \dfrac x {1 + \paren {x^2 + y}^2}
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \map {f_1} {x, y}
| r = \map {\dfrac \partial {\partial x} } {x \map \arctan {x^2 + y} }
| c = {{Defof|Partial Derivative}}
}}
{{eqn | r = \map \arctan {x^2 + y} \map {\dfrac \partial {\partial x} } x + x \map {\dfrac \partial {\partial x} } {\map \arctan {x^2 + y} }
| c = Pr... | Let:
:$\map f {x, y} = x \map \arctan {x^2 + y}$
Then:
{{begin-eqn}}
{{eqn | l = \map {f_1} {1, 0}
| r = \dfrac \pi 4 + 1
}}
{{eqn | l = \map {f_2} {x, y}
| r = \dfrac x {1 + \paren {x^2 + y}^2}
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \map {f_1} {x, y}
| r = \map {\dfrac \partial {\partial x} } {x \map \arctan {x^2 + y} }
| c = {{Defof|Partial Derivative}}
}}
{{eqn | r = \map \arctan {x^2 + y} \map {\dfrac \partial {\partial x} } x + x \map {\dfrac \partial {\partial x} } {\map \arctan {x^2 + y} }
| c = [[... | Partial Derivatives of x tan^-1 (x^2 + y) | https://proofwiki.org/wiki/Partial_Derivatives_of_x_tan^-1_(x^2_+_y) | https://proofwiki.org/wiki/Partial_Derivatives_of_x_tan^-1_(x^2_+_y) | [
"Examples of Partial Derivatives"
] | [] | [
"Product Rule for Derivatives",
"Product Rule for Derivatives",
"Derivative of Arctangent Function",
"Derivative of Composite Function",
"Tangent of 45 Degrees",
"Derivative of Identity Function",
"Derivative of Arctangent Function",
"Derivative of Composite Function"
] |
proofwiki-17807 | Partial Derivatives of x ln y^2 + y e^z | Let:
:$\map f {x, y, z} = x \ln y^2 + y e^z$
Then:
{{begin-eqn}}
{{eqn | l = \map {f_1} {1, -1, 0}
| r = 0
}}
{{eqn | l = \map {f_2} {x, x y, y + z}
| r = \dfrac 2 y + e^{y + z}
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \map {f_1} {x, y, z}
| r = \ln y^2
| c = Derivative of Constant Multiple, Derivative of Identity Function, holding $y$ and $z$ constant
}}
{{eqn | ll= \leadsto
| l = \map {f_1} {1, -1, 0}
| r = \map \ln {\paren {-1}^2}
| c = substituting $\tuple {1, -1, 0}$ for $\... | Let:
:$\map f {x, y, z} = x \ln y^2 + y e^z$
Then:
{{begin-eqn}}
{{eqn | l = \map {f_1} {1, -1, 0}
| r = 0
}}
{{eqn | l = \map {f_2} {x, x y, y + z}
| r = \dfrac 2 y + e^{y + z}
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \map {f_1} {x, y, z}
| r = \ln y^2
| c = [[Derivative of Constant Multiple]], [[Derivative of Identity Function]], holding $y$ and $z$ [[Definition:Constant|constant]]
}}
{{eqn | ll= \leadsto
| l = \map {f_1} {1, -1, 0}
| r = \map \ln {\paren {-1}^2}
| c = substit... | Partial Derivatives of x ln y^2 + y e^z | https://proofwiki.org/wiki/Partial_Derivatives_of_x_ln_y^2_+_y_e^z | https://proofwiki.org/wiki/Partial_Derivatives_of_x_ln_y^2_+_y_e^z | [
"Examples of Partial Derivatives"
] | [] | [
"Derivative of Constant Multiple",
"Derivative of Identity Function",
"Definition:Constant",
"Natural Logarithm of 1 is 0",
"Derivative of Natural Logarithm Function",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Definition:Constant"
] |
proofwiki-17808 | Partial Derivatives of x^y^z | Let:
:$u = x^{\paren {y^z} }$
Then:
{{begin-eqn}}
{{eqn | l = \dfrac {\partial u} {\partial x}
| r = y^z x^{\paren {y^z - 1} }
}}
{{eqn | l = \dfrac {\partial u} {\partial y}
| r = x^{y^z} z y^{z - 1} \ln x
}}
{{eqn | l = \dfrac {\partial u} {\partial z}
| r = x^{\paren {y^z} } y^z \ln x \ln y
}}
{{en... | {{begin-eqn}}
{{eqn | l = \dfrac {\partial u} {\partial x}
| r = y^z x^{\paren {y^z - 1} }
| c = Power Rule for Derivatives, keeping $y$ and $z$ constant
}}
{{end-eqn}}
{{qed|lemma}}
{{begin-eqn}}
{{eqn | l = u
| r = x^{\paren {y^z} }
| c =
}}
{{eqn | ll= \leadsto
| l = \ln u
| r = ... | Let:
:$u = x^{\paren {y^z} }$
Then:
{{begin-eqn}}
{{eqn | l = \dfrac {\partial u} {\partial x}
| r = y^z x^{\paren {y^z - 1} }
}}
{{eqn | l = \dfrac {\partial u} {\partial y}
| r = x^{y^z} z y^{z - 1} \ln x
}}
{{eqn | l = \dfrac {\partial u} {\partial z}
| r = x^{\paren {y^z} } y^z \ln x \ln y
}}
{{... | {{begin-eqn}}
{{eqn | l = \dfrac {\partial u} {\partial x}
| r = y^z x^{\paren {y^z - 1} }
| c = [[Power Rule for Derivatives]], keeping $y$ and $z$ [[Definition:Constant|constant]]
}}
{{end-eqn}}
{{qed|lemma}}
{{begin-eqn}}
{{eqn | l = u
| r = x^{\paren {y^z} }
| c =
}}
{{eqn | ll= \leadsto
... | Partial Derivatives of x^y^z | https://proofwiki.org/wiki/Partial_Derivatives_of_x^y^z | https://proofwiki.org/wiki/Partial_Derivatives_of_x^y^z | [
"Examples of Partial Derivatives"
] | [] | [
"Power Rule for Derivatives",
"Definition:Constant",
"Logarithm of Power",
"Derivative of Natural Logarithm Function",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Derivative of Natural Logarithm Function",
"Derivative of General Exponential Function",
"Derivative of Composite ... |
proofwiki-17809 | Partial Derivatives of x^u + u^y | Let:
:$u = x^u + u^y$
Then:
{{begin-eqn}}
{{eqn | l = \dfrac {\partial u} {\partial x}
| r = \frac {u^2} {x \paren {1 - u \ln x - y} }
}}
{{eqn | l = \dfrac {\partial u} {\partial y}
| r = \frac {u \ln u} {1 - u \ln x - y}
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = u
| r = x^u + u^y
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \ln u
| r = u \ln x + y \ln u
| c =
}}
{{eqn | ll= \leadsto
| l = \frac 1 u \dfrac {\partial u} {\partial x}
| r = \frac u x + \ln x \dfrac {\partial u} {\partial x} + \frac y u \dfr... | Let:
:$u = x^u + u^y$
Then:
{{begin-eqn}}
{{eqn | l = \dfrac {\partial u} {\partial x}
| r = \frac {u^2} {x \paren {1 - u \ln x - y} }
}}
{{eqn | l = \dfrac {\partial u} {\partial y}
| r = \frac {u \ln u} {1 - u \ln x - y}
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = u
| r = x^u + u^y
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \ln u
| r = u \ln x + y \ln u
| c =
}}
{{eqn | ll= \leadsto
| l = \frac 1 u \dfrac {\partial u} {\partial x}
| r = \frac u x + \ln x \dfrac {\partial u} {\partial x} + \frac y u \dfr... | Partial Derivatives of x^u + u^y | https://proofwiki.org/wiki/Partial_Derivatives_of_x^u_+_u^y | https://proofwiki.org/wiki/Partial_Derivatives_of_x^u_+_u^y | [
"Examples of Partial Derivatives"
] | [] | [
"Derivative of Natural Logarithm Function",
"Derivative of Composite Function",
"Derivative of Natural Logarithm Function",
"Product Rule for Derivatives",
"Derivative of Composite Function"
] |
proofwiki-17810 | Partial Derivatives of u^2 + x^2 + y^2, u - v^3 + 3 x | Let:
{{begin-eqn}}
{{eqn | l = u^2 + x^2 + y^2
| r = 3
}}
{{eqn | l = u - v^3 + 3 x
| r = 4
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \dfrac {\partial u} {\partial x}
| r = -\dfrac x u
}}
{{eqn | l = \dfrac {\partial u} {\partial y}
| r = -\dfrac y u
}}
{{eqn | l = \dfrac {\partial v} {\par... | {{begin-eqn}}
{{eqn | l = 2 u \dfrac {\partial u} {\partial x} + 2 x
| r = 0
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \dfrac {\partial u} {\partial x}
| r = -\dfrac x u
| c =
}}
{{end-eqn}}
{{qed|lemma}}
{{begin-eqn}}
{{eqn | l = 2 u \dfrac {\partial u} {\partial y} + 2 y
... | Let:
{{begin-eqn}}
{{eqn | l = u^2 + x^2 + y^2
| r = 3
}}
{{eqn | l = u - v^3 + 3 x
| r = 4
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \dfrac {\partial u} {\partial x}
| r = -\dfrac x u
}}
{{eqn | l = \dfrac {\partial u} {\partial y}
| r = -\dfrac y u
}}
{{eqn | l = \dfrac {\partial v} {\... | {{begin-eqn}}
{{eqn | l = 2 u \dfrac {\partial u} {\partial x} + 2 x
| r = 0
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \dfrac {\partial u} {\partial x}
| r = -\dfrac x u
| c =
}}
{{end-eqn}}
{{qed|lemma}}
{{begin-eqn}}
{{eqn | l = 2 u \dfrac {\partial u} {\partial y} + 2 y
... | Partial Derivatives of u^2 + x^2 + y^2, u - v^3 + 3 x | https://proofwiki.org/wiki/Partial_Derivatives_of_u^2_+_x^2_+_y^2,_u_-_v^3_+_3_x | https://proofwiki.org/wiki/Partial_Derivatives_of_u^2_+_x^2_+_y^2,_u_-_v^3_+_3_x | [
"Examples of Partial Derivatives"
] | [] | [] |
proofwiki-17811 | Third Partial Derivatives of x^y | Let:
:$u = x^y$
Then:
:$\dfrac {\partial^3 u} {\partial x^2 \partial y} = \dfrac {\partial^3 u} {\partial x \partial y \partial x}$ | {{begin-eqn}}
{{eqn | l = \dfrac {\partial u} {\partial y}
| r = x^y \ln x
| c = Derivative of General Logarithm Function keeping $x$ constant
}}
{{eqn | ll= \leadsto
| l = \dfrac {\partial^2 u} {\partial x \partial y}
| r = \map {\dfrac \partial {\partial x} } {\dfrac {\partial u} {\partial y} ... | Let:
:$u = x^y$
Then:
:$\dfrac {\partial^3 u} {\partial x^2 \partial y} = \dfrac {\partial^3 u} {\partial x \partial y \partial x}$ | {{begin-eqn}}
{{eqn | l = \dfrac {\partial u} {\partial y}
| r = x^y \ln x
| c = [[Derivative of General Logarithm Function]] keeping $x$ constant
}}
{{eqn | ll= \leadsto
| l = \dfrac {\partial^2 u} {\partial x \partial y}
| r = \map {\dfrac \partial {\partial x} } {\dfrac {\partial u} {\partial... | Third Partial Derivatives of x^y | https://proofwiki.org/wiki/Third_Partial_Derivatives_of_x^y | https://proofwiki.org/wiki/Third_Partial_Derivatives_of_x^y | [
"Examples of Partial Derivatives"
] | [] | [
"Derivative of General Logarithm Function",
"Power Rule for Derivatives",
"Derivative of Natural Logarithm Function",
"Product Rule for Derivatives",
"Derivative of Natural Logarithm Function",
"Power Rule for Derivatives",
"Power Rule for Derivatives",
"Product Rule for Derivatives",
"Derivative of... |
proofwiki-17812 | Number of Terms in Homogeneous Polynomial | The number of terms in a homogeneous polynomial of degree $n$ in $m$ indeterminates is given by:
:$N = \dbinom {n + m - 1} n = \dfrac {\paren {n + m - 1}!}{n! \, \paren {m - 1}!}$ | {{ProofWanted|By induction, probably, but really need to rationalise the existing material on polynomials}} | The number of [[Definition:Term of Polynomial|terms]] in a [[Definition:Homogeneous Polynomial|homogeneous polynomial]] of [[Definition:Degree of Polynomial|degree]] $n$ in $m$ [[Definition:Indeterminate|indeterminates]] is given by:
:$N = \dbinom {n + m - 1} n = \dfrac {\paren {n + m - 1}!}{n! \, \paren {m - 1}!}$ | {{ProofWanted|By induction, probably, but really need to rationalise the existing material on polynomials}} | Number of Terms in Homogeneous Polynomial | https://proofwiki.org/wiki/Number_of_Terms_in_Homogeneous_Polynomial | https://proofwiki.org/wiki/Number_of_Terms_in_Homogeneous_Polynomial | [
"Homogeneous Polynomials"
] | [
"Definition:Polynomial/Term",
"Definition:Homogeneous Polynomial",
"Definition:Degree of Polynomial",
"Definition:Indeterminate"
] | [] |
proofwiki-17813 | Heine-Borel Theorem/Normed Vector Space/Necessary Condition | Let $\struct {X, \norm {\, \cdot \,} }$ be a finite-dimensional normed vector space.
Let $K \subseteq X$ be closed and bounded.
Then $K$ is a compact subset. | Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $K$.
$\sequence {x_n}_{n \mathop \in \N}$ is bounded.
We have that bounded sequence in finite-dimensional space has a convergent subsequence.
Hence, $\sequence {x_n}_{n \mathop \in \N}$ has a convergent subsequence $\sequence {x_{n_k} }_{k \mathop \in \N}$.
Deno... | Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Finite Dimensional Vector Space|finite-dimensional]] [[Definition:Normed Vector Space|normed vector space]].
Let $K \subseteq X$ be [[Definition:Closed Set in Normed Vector Space|closed]] and [[Definition:Bounded Subset of Normed Vector Space|bounded]].
Then ... | Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence in $K$]].
$\sequence {x_n}_{n \mathop \in \N}$ is [[Definition:Bounded Sequence in Normed Vector Space|bounded]].
We have that [[Bounded Sequence in Euclidean Space has Convergent Subsequence/Proof 3|bounded sequence in finite-dimensional sp... | Heine-Borel Theorem/Normed Vector Space/Necessary Condition | https://proofwiki.org/wiki/Heine-Borel_Theorem/Normed_Vector_Space/Necessary_Condition | https://proofwiki.org/wiki/Heine-Borel_Theorem/Normed_Vector_Space/Necessary_Condition | [
"Heine-Borel Theorem"
] | [
"Definition:Dimension of Vector Space/Finite",
"Definition:Normed Vector Space",
"Definition:Closed Set/Normed Vector Space",
"Definition:Bounded Subset of Normed Vector Space",
"Definition:Compact Space/Normed Vector Space/Subspace",
"Definition:Subset"
] | [
"Definition:Sequence",
"Definition:Bounded Sequence/Normed Vector Space",
"Bounded Sequence in Euclidean Space has Convergent Subsequence/Proof 3",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Subsequence",
"Definition:Limit of Sequence/Normed Vector Space",
"Definition:Assumption",... |
proofwiki-17814 | Compact Subset of Normed Vector Space is Closed and Bounded | Let $\struct {X, \norm {\,\cdot\,}}$ be a normed vector space.
Let $K \subset X$ be a compact subset.
Then $K$ is closed and bounded. | === Closedness ===
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $K$.
Suppose, $\sequence {x_n}_{n \mathop \in \N}$ converges to $L \in K$.
Then there is a subsequence $\sequence {x_{n_k}}_{k \mathop \in \N}$ convergent to $L' \in K$.
But $\sequence {x_{n_k}}_{k \mathop \in \N}$ is a subsequence of $\sequ... | Let $\struct {X, \norm {\,\cdot\,}}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $K \subset X$ be a [[Definition:Compact Space/Normed Vector Space/Subspace|compact]] [[Definition:Subset|subset]].
Then $K$ is [[Definition:Closed Set of Normed Vector Space|closed]] and [[Definition:Bounded Subset ... | === Closedness ===
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence in $K$]].
Suppose, $\sequence {x_n}_{n \mathop \in \N}$ [[Definition:Convergent Sequence in Normed Vector Space|converges]] to $L \in K$.
Then there is a [[Definition:Subsequence|subsequence]] $\sequence {x_{n_k}}_{k \ma... | Compact Subset of Normed Vector Space is Closed and Bounded | https://proofwiki.org/wiki/Compact_Subset_of_Normed_Vector_Space_is_Closed_and_Bounded | https://proofwiki.org/wiki/Compact_Subset_of_Normed_Vector_Space_is_Closed_and_Bounded | [
"Boundedness",
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Compact Space/Normed Vector Space/Subspace",
"Definition:Subset",
"Definition:Closed Set/Normed Vector Space",
"Definition:Bounded Subset of Normed Vector Space"
] | [
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Subsequence",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Subsequence",
"Limit of Subsequence equals Limit of Sequence/Normed Vector Space",
"Convergent Sequence in Normed Vector Space has Uniq... |
proofwiki-17815 | Sine of X over X is not Continuous at 0 | Let $f$ be the real function defined as:
:$\map f x := \dfrac {\sin x} x$
Then $f$ is not continuous at $x = 0$. | For $f$ to be continuous at $x = 0$ it is necessary that it be defined there.
But at the point $x = 0$, we have that $\map f x = \dfrac {\sin 0} 0$.
Division by $0$ is not defined.
Hence $f$ is not continuous at $x = 0$.
{{qed}} | Let $f$ be the [[Definition:Real Function|real function]] defined as:
:$\map f x := \dfrac {\sin x} x$
Then $f$ is not [[Definition:Continuous Real Function|continuous]] at $x = 0$. | For $f$ to be [[Definition:Continuous Real Function|continuous]] at $x = 0$ it is necessary that it be defined there.
But at the point $x = 0$, we have that $\map f x = \dfrac {\sin 0} 0$.
[[Definition:Real Division|Division]] by $0$ is not defined.
Hence $f$ is not [[Definition:Continuous Real Function|continuous]]... | Sine of X over X is not Continuous at 0 | https://proofwiki.org/wiki/Sine_of_X_over_X_is_not_Continuous_at_0 | https://proofwiki.org/wiki/Sine_of_X_over_X_is_not_Continuous_at_0 | [
"Examples of Continuous Real Functions"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function"
] | [
"Definition:Continuous Real Function",
"Definition:Division/Field/Real Numbers",
"Definition:Continuous Real Function"
] |
proofwiki-17816 | Reciprocal Function is Discontinuous at Zero | Let $f$ be the real function defined as:
:$\map f x = \dfrac 1 x$
for $x \ne 0$
Then $\map f x$ cannot be extended to a real function $g$ on $\R$ such that $\map g x$ is continuous on $\openint {-1} 1$. | It is apparent that $f$ is itself not continuous at $x = 0$.
We need to show that whatever we define $\map g 0$ to be, $\ds \lim_{x \mathop \to 0} \map f x \ne \map f 0$.
Let $\map g 0 = c$.
Let $\epsilon$ be selected.
Let $x$ be chosen such that $x < \dfrac 1 c$.
Then $\dfrac 1 x > c$.
It follows that
:$\ds \lim_{x \m... | Let $f$ be the [[Definition:Real Function|real function]] defined as:
:$\map f x = \dfrac 1 x$
for $x \ne 0$
Then $\map f x$ cannot be [[Definition:Extension of Mapping|extended]] to a [[Definition:Real Function|real function]] $g$ on $\R$ such that $\map g x$ is [[Definition:Continuous on Interval|continuous]] on $\... | It is apparent that $f$ is itself not [[Definition:Continuous Real Function at Point|continuous]] at $x = 0$.
We need to show that whatever we define $\map g 0$ to be, $\ds \lim_{x \mathop \to 0} \map f x \ne \map f 0$.
Let $\map g 0 = c$.
Let $\epsilon$ be selected.
Let $x$ be chosen such that $x < \dfrac 1 c$.
... | Reciprocal Function is Discontinuous at Zero | https://proofwiki.org/wiki/Reciprocal_Function_is_Discontinuous_at_Zero | https://proofwiki.org/wiki/Reciprocal_Function_is_Discontinuous_at_Zero | [
"Reciprocals"
] | [
"Definition:Real Function",
"Definition:Extension of Mapping",
"Definition:Real Function",
"Definition:Continuous Real Function/Interval"
] | [
"Definition:Continuous Real Function/Point"
] |
proofwiki-17817 | Right-Hand Derivative not Limit of Derivative from Right | Let $f$ be a real function.
Let the right-hand derivative $f'_+$ of $f$ exist.
Then it is not necessarily the case that:
:$\map {f'_+} a$
is the same thing as:
:$\map {f'} {a^+}$ | By definition:
:$\map {f'_+} a := \ds \lim_{h \mathop \to 0} \dfrac {\map f {a + h} - \map f a} h$
while:
:$\map {f'} {a^+} := \ds \lim_{x \mathop \to a^+} \map {f'} x$
Let:
:$\map f x = \begin {cases} x^2 \sin \dfrac 1 x & : x \ne 0 \\ 0 & : x = 0 \end {cases}$.
Then:
{{begin-eqn}}
{{eqn | l = \map {f'_+} 0
| r ... | Let $f$ be a [[Definition:Real Function|real function]].
Let the [[Definition:Real Right-Hand Derivative|right-hand derivative]] $f'_+$ of $f$ exist.
Then it is not necessarily the case that:
:$\map {f'_+} a$
is the same thing as:
:$\map {f'} {a^+}$ | By definition:
:$\map {f'_+} a := \ds \lim_{h \mathop \to 0} \dfrac {\map f {a + h} - \map f a} h$
while:
:$\map {f'} {a^+} := \ds \lim_{x \mathop \to a^+} \map {f'} x$
Let:
:$\map f x = \begin {cases} x^2 \sin \dfrac 1 x & : x \ne 0 \\ 0 & : x = 0 \end {cases}$.
Then:
{{begin-eqn}}
{{eqn | l = \map {f'_+} 0
... | Right-Hand Derivative not Limit of Derivative from Right | https://proofwiki.org/wiki/Right-Hand_Derivative_not_Limit_of_Derivative_from_Right | https://proofwiki.org/wiki/Right-Hand_Derivative_not_Limit_of_Derivative_from_Right | [
"Differential Calculus"
] | [
"Definition:Real Function",
"Definition:Right-Hand Derivative/Real Function"
] | [
"Limit of Real Function/Examples/x times Sine of Reciprocal of x at 0"
] |
proofwiki-17818 | Function in Differentiability Class 1 is also in Continuity Class | Let $f$ be a real function.
Let $f$ be an element of differentiability class $C^1$.
Then $f$ is also an element of the class $C$ of continuous real functions. | By definition of $C^1$, $f \in C^1$ {{iff}} $f$ is differentiable.
By definition of $C$, $f \in C$ {{iff}} $f$ is continuous.
The result follows from Differentiable Function is Continuous.
{{qed}} | Let $f$ be a [[Definition:Real Function|real function]].
Let $f$ be an [[Definition:Element|element]] of [[Definition:Differentiability Class|differentiability class $C^1$]].
Then $f$ is also an [[Definition:Element|element]] of the [[Definition:Class of Continuous Real Functions|class $C$ of continuous real functio... | By definition of $C^1$, $f \in C^1$ {{iff}} $f$ is [[Definition:Differentiable Real Function|differentiable]].
By definition of $C$, $f \in C$ {{iff}} $f$ is [[Definition:Continuous Real Function|continuous]].
The result follows from [[Differentiable Function is Continuous]].
{{qed}} | Function in Differentiability Class 1 is also in Continuity Class | https://proofwiki.org/wiki/Function_in_Differentiability_Class_1_is_also_in_Continuity_Class | https://proofwiki.org/wiki/Function_in_Differentiability_Class_1_is_also_in_Continuity_Class | [
"Differentiability Classes"
] | [
"Definition:Real Function",
"Definition:Element",
"Definition:Differentiability Class",
"Definition:Element",
"Definition:Continuous Real Function/Class"
] | [
"Definition:Differentiable Mapping/Real Function",
"Definition:Continuous Real Function",
"Differentiable Function is Continuous"
] |
proofwiki-17819 | Differentiability Class is Subset of Differentiability Class of Lower Order | Let $f$ be a real function.
Let $f$ be an element of differentiability class $C^n$.
Then:
:$\forall k \in \set {0, 1, \ldots, n - 1}: f \in C^k$
That is, $f$ is in all differentiability classes of order less than $n$. | The proof proceeds by induction.
For all $n \in \Z_{\ge 1}$, let $\map P n$ be the proposition:
:$f \in C^n \implies \forall k \in \set {0, 1, \ldots, n - 1}: f \in C^k$ | Let $f$ be a [[Definition:Real Function|real function]].
Let $f$ be an [[Definition:Element|element]] of [[Definition:Differentiability Class|differentiability class $C^n$]].
Then:
:$\forall k \in \set {0, 1, \ldots, n - 1}: f \in C^k$
That is, $f$ is in all [[Definition:Differentiability Class|differentiability cl... | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{\ge 1}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$f \in C^n \implies \forall k \in \set {0, 1, \ldots, n - 1}: f \in C^k$ | Differentiability Class is Subset of Differentiability Class of Lower Order | https://proofwiki.org/wiki/Differentiability_Class_is_Subset_of_Differentiability_Class_of_Lower_Order | https://proofwiki.org/wiki/Differentiability_Class_is_Subset_of_Differentiability_Class_of_Lower_Order | [
"Differentiability Classes"
] | [
"Definition:Real Function",
"Definition:Element",
"Definition:Differentiability Class",
"Definition:Differentiability Class",
"Definition:Derivative/Higher Derivatives/Order of Derivative"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-17820 | Niven's Theorem/Lemma | For any integer $n \ge 1$, there exists a polynomial $\map {F_n} x$ such that:
:$\map {F_n} {2 \cos t} = 2 \cos n t$
In addition:
:$\deg F_n = n$
and $F_n$ is a monic polynomial with integer coefficients. | The proof proceeds by induction.
For $n = 1$, it is seen that:
:$\map {F_1} x = x$
fulfils the propositions.
For $n = 2$:
:$\map {F_2} x = x^2 - 2$
For $n > 2$:
{{begin-eqn}}
{{eqn | l = 2 \map \cos {n - 1} t \cos t
| r = \cos n t + \map \cos {n - 2} t
| c =
}}
{{eqn | ll= \leadsto
| l = 2 \cos n t
... | For any [[Definition:Integer|integer]] $n \ge 1$, there exists a [[Definition:Polynomial over Real Numbers|polynomial]] $\map {F_n} x$ such that:
:$\map {F_n} {2 \cos t} = 2 \cos n t$
In addition:
:$\deg F_n = n$
and $F_n$ is a [[Definition:Monic Polynomial|monic polynomial]] with [[Definition:Integer|integer]] [[Defi... | The proof proceeds by [[Proof by Mathematical Induction|induction]].
For $n = 1$, it is seen that:
:$\map {F_1} x = x$
fulfils the propositions.
For $n = 2$:
:$\map {F_2} x = x^2 - 2$
For $n > 2$:
{{begin-eqn}}
{{eqn | l = 2 \map \cos {n - 1} t \cos t
| r = \cos n t + \map \cos {n - 2} t
| c =
}}
{{eqn... | Niven's Theorem/Lemma | https://proofwiki.org/wiki/Niven's_Theorem/Lemma | https://proofwiki.org/wiki/Niven's_Theorem/Lemma | [
"Proofs by Induction",
"Niven's Theorem"
] | [
"Definition:Integer",
"Definition:Polynomial/Real Numbers",
"Definition:Monic Polynomial",
"Definition:Integer",
"Definition:Coefficient of Polynomial"
] | [
"Principle of Mathematical Induction",
"Definition:Leading Coefficient of Polynomial",
"Definition:Leading Coefficient of Polynomial",
"Category:Proofs by Induction",
"Category:Niven's Theorem"
] |
proofwiki-17821 | Closed and Bounded Subset of Normed Vector Space is not necessarily Compact | Let $\struct {X, \norm {\,\cdot\,} }$ be a normed vector space.
Let $K \subset X$ be a closed and bounded subset.
Then $K$ is not necessarily compact. | Let $\struct {\ell^2, \norm {\, \cdot \,}}_2$ be the 2-sequence space.
Let $K$ be a closed unit ball in $\struct {\ell^2, \norm {\, \cdot \,}}_2$:
:$K := \set {\mathbf x \in \ell^2 : \norm {\mathbf x}_2 \le 1}$
$K$ is bounded and closed.
Let $\sequence {\mathbf x_n}_{n \mathop \in \N}$ be a sequence such that:
:$\mathb... | Let $\struct {X, \norm {\,\cdot\,} }$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $K \subset X$ be a [[Definition:Closed Set of Normed Vector Space|closed]] and [[Definition:Bounded Subset of Normed Vector Space|bounded]] [[Definition:Subset|subset]].
Then $K$ is not necessarily [[Definition:Com... | Let $\struct {\ell^2, \norm {\, \cdot \,}}_2$ be the [[P-Sequence Space with P-Norm forms Normed Vector Space|2-sequence space]].
Let $K$ be a [[Definition:Closed Ball in Normed Vector Space|closed]] [[Definition:Unit Ball|unit ball]] in $\struct {\ell^2, \norm {\, \cdot \,}}_2$:
:$K := \set {\mathbf x \in \ell^2 : \... | Closed and Bounded Subset of Normed Vector Space is not necessarily Compact/Proof 1 | https://proofwiki.org/wiki/Closed_and_Bounded_Subset_of_Normed_Vector_Space_is_not_necessarily_Compact | https://proofwiki.org/wiki/Closed_and_Bounded_Subset_of_Normed_Vector_Space_is_not_necessarily_Compact/Proof_1 | [
"Closed and Bounded Subset of Normed Vector Space is not necessarily Compact",
"Compact Normed Vector Spaces",
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Closed Set/Normed Vector Space",
"Definition:Bounded Subset of Normed Vector Space",
"Definition:Subset",
"Definition:Compact Space/Normed Vector Space/Subspace"
] | [
"P-Sequence Space with P-Norm forms Normed Vector Space",
"Definition:Closed Ball/Normed Vector Space",
"Definition:Unit Ball",
"Definition:Bounded Subset of Normed Vector Space/Definition 1",
"Closed Ball is Closed/Normed Vector Space",
"Definition:Sequence",
"Definition:Vector Quantity/Component",
"... |
proofwiki-17822 | Closed and Bounded Subset of Normed Vector Space is not necessarily Compact | Let $\struct {X, \norm {\,\cdot\,} }$ be a normed vector space.
Let $K \subset X$ be a closed and bounded subset.
Then $K$ is not necessarily compact. | Let $\struct {\CC \closedint 0 1, \norm {\, \cdot \,}_\infty}$ be the normed vector space of continuous on closed interval real-valued functions with supremum norm.
Consider a closed unit ball with the center at $0$ at $\struct {\CC, \norm {\, \cdot \,}_\infty}$:
:$K := \set {x \in \CC \closedint 0 1 : \norm x_\infty \... | Let $\struct {X, \norm {\,\cdot\,} }$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $K \subset X$ be a [[Definition:Closed Set of Normed Vector Space|closed]] and [[Definition:Bounded Subset of Normed Vector Space|bounded]] [[Definition:Subset|subset]].
Then $K$ is not necessarily [[Definition:Com... | Let $\struct {\CC \closedint 0 1, \norm {\, \cdot \,}_\infty}$ be the [[Space of Continuous on Closed Interval Real-Valued Functions with Supremum Norm forms Normed Vector Space|normed vector space of continuous on closed interval real-valued functions with supremum norm]].
Consider a [[Definition:Closed Unit Ball|clo... | Closed and Bounded Subset of Normed Vector Space is not necessarily Compact/Proof 2 | https://proofwiki.org/wiki/Closed_and_Bounded_Subset_of_Normed_Vector_Space_is_not_necessarily_Compact | https://proofwiki.org/wiki/Closed_and_Bounded_Subset_of_Normed_Vector_Space_is_not_necessarily_Compact/Proof_2 | [
"Closed and Bounded Subset of Normed Vector Space is not necessarily Compact",
"Compact Normed Vector Spaces",
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Closed Set/Normed Vector Space",
"Definition:Bounded Subset of Normed Vector Space",
"Definition:Subset",
"Definition:Compact Space/Normed Vector Space/Subspace"
] | [
"Space of Continuous on Closed Interval Real-Valued Functions with Supremum Norm forms Normed Vector Space",
"Definition:Closed Unit Ball",
"Definition:Open Ball/Center",
"Definition:Bounded Subset of Normed Vector Space",
"Closed Real Interval is Closed Set",
"Definition:Natural Numbers",
"Definition:C... |
proofwiki-17823 | Floor Function is Continuous on Right at Integer | Let $f$ be the real function defined as:
:$\map f x = \floor x$
where $\floor{\, \cdot \,}$ denotes the floor function.
Let $n \in \Z$ be an integer.
Then $\map f x$ is continuous on the right at $n$. | From Real Number is Integer iff equals Floor:
:$\floor n = n$
By definition $\floor x$ is the unique integer such that:
:$\floor x \le x < \floor x + 1$
Consider the open real interval:
:$\II = \openint n {n + 1}$
By definition of $\floor x$ we have that:
:$\forall x \in \II: \floor x = n$
That is:
:$\forall x \in \II:... | Let $f$ be the [[Definition:Real Function|real function]] defined as:
:$\map f x = \floor x$
where $\floor{\, \cdot \,}$ denotes the [[Definition:Floor Function|floor function]].
Let $n \in \Z$ be an [[Definition:Integer|integer]].
Then $\map f x$ is [[Definition:Right-Continuous at Point|continuous on the right]] a... | From [[Real Number is Integer iff equals Floor]]:
:$\floor n = n$
By definition $\floor x$ is the [[Definition:Unique|unique]] [[Definition:Integer|integer]] such that:
:$\floor x \le x < \floor x + 1$
Consider the [[Definition:Open Real Interval|open real interval]]:
:$\II = \openint n {n + 1}$
By definition of $\f... | Floor Function is Continuous on Right at Integer | https://proofwiki.org/wiki/Floor_Function_is_Continuous_on_Right_at_Integer | https://proofwiki.org/wiki/Floor_Function_is_Continuous_on_Right_at_Integer | [
"Floor Function",
"Examples of One-Sided Continuity"
] | [
"Definition:Real Function",
"Definition:Floor Function",
"Definition:Integer",
"Definition:Continuous Real Function/Right-Continuous"
] | [
"Real Number is Integer iff equals Floor",
"Definition:Unique",
"Definition:Integer",
"Definition:Real Interval/Open",
"Definition:Continuous Real Function/Right-Continuous"
] |
proofwiki-17824 | Floor Function is Not Continuous on Left at Integer | Let $f$ be the real function defined as:
:$\map f x = \floor x$
where $\floor{\, \cdot \,}$ denotes the floor function.
Let $n \in \Z$ be an integer.
Then $\map f x$ is not continuous on the left at $n$. | From Real Number is Integer iff equals Floor:
:$\floor n = n$
By definition $\floor x$ is the unique integer such that:
:$\floor x \le x < \floor x + 1$
Consider the open real interval:
:$\II = \openint {n - 1} n$
By definition of $\floor x$ we have that:
:$\forall x \in \II: \floor x = n - 1$
That is:
:$\forall x \in ... | Let $f$ be the [[Definition:Real Function|real function]] defined as:
:$\map f x = \floor x$
where $\floor{\, \cdot \,}$ denotes the [[Definition:Floor Function|floor function]].
Let $n \in \Z$ be an [[Definition:Integer|integer]].
Then $\map f x$ is not [[Definition:Left-Continuous at Point|continuous on the left]]... | From [[Real Number is Integer iff equals Floor]]:
:$\floor n = n$
By definition $\floor x$ is the [[Definition:Unique|unique]] [[Definition:Integer|integer]] such that:
:$\floor x \le x < \floor x + 1$
Consider the [[Definition:Open Real Interval|open real interval]]:
:$\II = \openint {n - 1} n$
By definition of $\f... | Floor Function is Not Continuous on Left at Integer | https://proofwiki.org/wiki/Floor_Function_is_Not_Continuous_on_Left_at_Integer | https://proofwiki.org/wiki/Floor_Function_is_Not_Continuous_on_Left_at_Integer | [
"Floor Function",
"Examples of One-Sided Continuity"
] | [
"Definition:Real Function",
"Definition:Floor Function",
"Definition:Integer",
"Definition:Continuous Real Function/Left-Continuous"
] | [
"Real Number is Integer iff equals Floor",
"Definition:Unique",
"Definition:Integer",
"Definition:Real Interval/Open",
"Definition:Continuous Real Function/Left-Continuous"
] |
proofwiki-17825 | Cauchy's Convergence Criterion/Real Numbers/Necessary Condition | Let $\sequence {x_n}$ be a sequence in $\R$.
Let $\sequence {x_n}$ be convergent.
Then $\sequence {x_n}$ is a Cauchy sequence. | Let $\sequence {x_n}$ be convergent.
Let $\struct {\R, d}$ be the metric space formed from $\R$ and the usual (Euclidean) metric:
:$\map d {x_1, x_2} = \size {x_1 - x_2}$
where $\size x$ is the absolute value of $x$.
This is proven to be a metric space in Real Number Line is Metric Space.
From Convergent Sequence in Me... | Let $\sequence {x_n}$ be a [[Definition:Real Sequence|sequence in $\R$]].
Let $\sequence {x_n}$ be [[Definition:Convergent Real Sequence|convergent]].
Then $\sequence {x_n}$ is a [[Definition:Real Cauchy Sequence|Cauchy sequence]]. | Let $\sequence {x_n}$ be [[Definition:Convergent Sequence|convergent]].
Let $\struct {\R, d}$ be the [[Definition:Metric Space|metric space]] formed from $\R$ and the [[Definition:Euclidean Metric on Real Vector Space|usual (Euclidean) metric]]:
:$\map d {x_1, x_2} = \size {x_1 - x_2}$
where $\size x$ is the [[Definit... | Cauchy's Convergence Criterion/Real Numbers/Necessary Condition/Proof 1 | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion/Real_Numbers/Necessary_Condition | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion/Real_Numbers/Necessary_Condition/Proof_1 | [
"Cauchy's Convergence Criterion"
] | [
"Definition:Real Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Cauchy Sequence/Real Numbers"
] | [
"Definition:Convergent Sequence",
"Definition:Metric Space",
"Definition:Euclidean Metric/Real Vector Space",
"Definition:Absolute Value",
"Definition:Metric Space",
"Real Number Line is Metric Space",
"Convergent Sequence is Cauchy Sequence/Metric Space",
"Definition:Convergent Sequence",
"Definiti... |
proofwiki-17826 | Cauchy's Convergence Criterion/Real Numbers/Necessary Condition | Let $\sequence {x_n}$ be a sequence in $\R$.
Let $\sequence {x_n}$ be convergent.
Then $\sequence {x_n}$ is a Cauchy sequence. | Let $\sequence {x_n}$ be a sequence in $\R$ that converges to the limit $l \in \R$.
Let $\epsilon > 0$.
Then also $\dfrac \epsilon 2 > 0$.
Because $\sequence {x_n}$ converges to $l$, we have:
:$\exists N: \forall n > N: \size {x_n - l} < \dfrac \epsilon 2$
So if $m > N$ and $n > N$, then:
{{begin-eqn}}
{{eqn | l = \si... | Let $\sequence {x_n}$ be a [[Definition:Real Sequence|sequence in $\R$]].
Let $\sequence {x_n}$ be [[Definition:Convergent Real Sequence|convergent]].
Then $\sequence {x_n}$ is a [[Definition:Real Cauchy Sequence|Cauchy sequence]]. | Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence]] in $\R$ that [[Definition:Convergent Real Sequence|converges]] to the [[Definition:Limit of Real Sequence|limit]] $l \in \R$.
Let $\epsilon > 0$.
Then also $\dfrac \epsilon 2 > 0$.
Because $\sequence {x_n}$ [[Definition:Convergent Real Sequence|converges]]... | Cauchy's Convergence Criterion/Real Numbers/Necessary Condition/Proof 2 | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion/Real_Numbers/Necessary_Condition | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion/Real_Numbers/Necessary_Condition/Proof_2 | [
"Cauchy's Convergence Criterion"
] | [
"Definition:Real Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Cauchy Sequence/Real Numbers"
] | [
"Definition:Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Limit of Sequence/Real Numbers",
"Definition:Convergent Sequence/Real Numbers",
"Triangle Inequality",
"Definition:Cauchy Sequence/Real Numbers"
] |
proofwiki-17827 | Cauchy's Convergence Criterion/Real Numbers/Sufficient Condition | Let $\sequence {x_n}$ be a sequence in $\R$.
Let $\sequence {x_n}$ be a Cauchy sequence.
Then $\sequence {x_n}$ is convergent. | Let $\sequence {a_n}$ be a Cauchy sequence in $\R$.
We have the result Real Number Line is Metric Space.
Hence by Convergent Subsequence of Cauchy Sequence, it is sufficient to show that $\sequence {a_n}$ has a convergent subsequence.
Since $\sequence {a_n}$ is Cauchy, by Real Cauchy Sequence is Bounded, it is also bou... | Let $\sequence {x_n}$ be a [[Definition:Real Sequence|sequence in $\R$]].
Let $\sequence {x_n}$ be a [[Definition:Real Cauchy Sequence|Cauchy sequence]].
Then $\sequence {x_n}$ is [[Definition:Convergent Real Sequence|convergent]]. | Let $\sequence {a_n}$ be a [[Definition:Real Cauchy Sequence|Cauchy sequence in $\R$]].
We have the result [[Real Number Line is Metric Space]].
Hence by [[Convergent Subsequence of Cauchy Sequence/Metric Space|Convergent Subsequence of Cauchy Sequence]], it is sufficient to show that $\sequence {a_n}$ has a [[Defini... | Cauchy's Convergence Criterion/Real Numbers/Sufficient Condition/Proof 1 | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion/Real_Numbers/Sufficient_Condition | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion/Real_Numbers/Sufficient_Condition/Proof_1 | [
"Cauchy's Convergence Criterion"
] | [
"Definition:Real Sequence",
"Definition:Cauchy Sequence/Real Numbers",
"Definition:Convergent Sequence/Real Numbers"
] | [
"Definition:Cauchy Sequence/Real Numbers",
"Real Number Line is Metric Space",
"Convergent Subsequence of Cauchy Sequence/Metric Space",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Subsequence",
"Definition:Cauchy Sequence/Real Numbers",
"Cauchy Sequence is Bounded/Real Numbers",
"Defin... |
proofwiki-17828 | Cauchy's Convergence Criterion/Real Numbers/Sufficient Condition | Let $\sequence {x_n}$ be a sequence in $\R$.
Let $\sequence {x_n}$ be a Cauchy sequence.
Then $\sequence {x_n}$ is convergent. | Let $\sequence {a_n}$ be a Cauchy sequence in $\R$.
By Real Cauchy Sequence is Bounded, $\sequence {a_n}$ is bounded.
By the Bolzano-Weierstrass Theorem, $\sequence {a_n}$ has a convergent subsequence $\sequence {a_{n_r} }$.
Let $a_{n_r} \to l$ as $r \to \infty$.
It is to be shown that $a_n \to l$ as $n \to \infty$.
Le... | Let $\sequence {x_n}$ be a [[Definition:Real Sequence|sequence in $\R$]].
Let $\sequence {x_n}$ be a [[Definition:Real Cauchy Sequence|Cauchy sequence]].
Then $\sequence {x_n}$ is [[Definition:Convergent Real Sequence|convergent]]. | Let $\sequence {a_n}$ be a [[Definition:Real Cauchy Sequence|Cauchy sequence in $\R$]].
By [[Real Cauchy Sequence is Bounded]], $\sequence {a_n}$ is [[Definition:Bounded Real Sequence|bounded]].
By the [[Bolzano-Weierstrass Theorem]], $\sequence {a_n}$ has a [[Definition:Convergent Real Sequence|convergent]] [[Defini... | Cauchy's Convergence Criterion/Real Numbers/Sufficient Condition/Proof 2 | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion/Real_Numbers/Sufficient_Condition | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion/Real_Numbers/Sufficient_Condition/Proof_2 | [
"Cauchy's Convergence Criterion"
] | [
"Definition:Real Sequence",
"Definition:Cauchy Sequence/Real Numbers",
"Definition:Convergent Sequence/Real Numbers"
] | [
"Definition:Cauchy Sequence/Real Numbers",
"Cauchy Sequence is Bounded/Real Numbers",
"Definition:Bounded Sequence/Real",
"Bolzano-Weierstrass Theorem",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Subsequence",
"Definition:Strictly Positive/Real Number",
"Definition:Cauchy Sequence/Real... |
proofwiki-17829 | Cauchy's Convergence Criterion/Real Numbers/Sufficient Condition | Let $\sequence {x_n}$ be a sequence in $\R$.
Let $\sequence {x_n}$ be a Cauchy sequence.
Then $\sequence {x_n}$ is convergent. | The aim is to define two sequences whose elements are respectively upper and lower bounds to subsequences of the sequence $\sequence {a_n}$.
It is then shown that these two sequences converge to the same limit.
This is used to prove that $\sequence {a_n}$ converges.
A sequence $\sequence {\epsilon_i}$ is introduced tha... | Let $\sequence {x_n}$ be a [[Definition:Real Sequence|sequence in $\R$]].
Let $\sequence {x_n}$ be a [[Definition:Real Cauchy Sequence|Cauchy sequence]].
Then $\sequence {x_n}$ is [[Definition:Convergent Real Sequence|convergent]]. | The aim is to define two [[Definition:Real Sequence|sequences]] whose [[Definition:Sequence|elements]] are respectively [[Definition:Upper Bound of Real Sequence|upper]] and [[Definition:Lower Bound of Real Sequence|lower bounds]] to [[Definition:Subsequence|subsequences]] of the [[Definition:Real Sequence|sequence]] $... | Cauchy's Convergence Criterion/Real Numbers/Sufficient Condition/Proof 3 | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion/Real_Numbers/Sufficient_Condition | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion/Real_Numbers/Sufficient_Condition/Proof_3 | [
"Cauchy's Convergence Criterion"
] | [
"Definition:Real Sequence",
"Definition:Cauchy Sequence/Real Numbers",
"Definition:Convergent Sequence/Real Numbers"
] | [
"Definition:Real Sequence",
"Definition:Sequence",
"Definition:Upper Bound of Sequence/Real",
"Definition:Lower Bound of Sequence/Real",
"Definition:Subsequence",
"Definition:Real Sequence",
"Definition:Real Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Limit of Sequence/Real... |
proofwiki-17830 | Cauchy's Convergence Criterion/Real Numbers/Sufficient Condition | Let $\sequence {x_n}$ be a sequence in $\R$.
Let $\sequence {x_n}$ be a Cauchy sequence.
Then $\sequence {x_n}$ is convergent. | Let $\sequence {a_n}$ be a Cauchy sequence in $\R$.
By Real Cauchy Sequence is Bounded, $\sequence {a_n}$ is bounded.
Thus $\sequence {a_n}$ is both bounded above and bounded below.
Let us create a monotone subsequence $\sequence {b_n}$ of $\sequence {a_n}$ using the following construction:
For each $m \in \N$, let $S_... | Let $\sequence {x_n}$ be a [[Definition:Real Sequence|sequence in $\R$]].
Let $\sequence {x_n}$ be a [[Definition:Real Cauchy Sequence|Cauchy sequence]].
Then $\sequence {x_n}$ is [[Definition:Convergent Real Sequence|convergent]]. | Let $\sequence {a_n}$ be a [[Definition:Real Cauchy Sequence|Cauchy sequence in $\R$]].
By [[Real Cauchy Sequence is Bounded]], $\sequence {a_n}$ is [[Definition:Bounded Real Sequence|bounded]].
Thus $\sequence {a_n}$ is both [[Definition:Bounded Above Real Sequence|bounded above]] and [[Definition:Bounded Below Real... | Cauchy's Convergence Criterion/Real Numbers/Sufficient Condition/Proof 4 | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion/Real_Numbers/Sufficient_Condition | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion/Real_Numbers/Sufficient_Condition/Proof_4 | [
"Cauchy's Convergence Criterion"
] | [
"Definition:Real Sequence",
"Definition:Cauchy Sequence/Real Numbers",
"Definition:Convergent Sequence/Real Numbers"
] | [
"Definition:Cauchy Sequence/Real Numbers",
"Cauchy Sequence is Bounded/Real Numbers",
"Definition:Bounded Sequence/Real",
"Definition:Bounded Above Sequence/Real",
"Definition:Bounded Below Sequence/Real",
"Definition:Monotone (Order Theory)/Sequence/Real Sequence",
"Definition:Subsequence",
"Definiti... |
proofwiki-17831 | Real Number Line is Banach Space | The real number line, along with the absolute value function, forms a Banach space over $\R$. | From Real Numbers form Vector Space we have the real number line $\R$ is a vector space over $\R$.
That the norm axioms are satisfied by the absolute value function is proven in Absolute Value is Norm.
Then we have Real Number Line is Complete Metric Space.
Hence the result.
{{qed}}
Category:Banach Spaces
Category:Rea... | The [[Definition:Real Number Line|real number line]], along with the [[Definition:Absolute Value|absolute value function]], forms a [[Definition:Banach Space|Banach space]] over $\R$. | From [[Real Numbers form Vector Space]] we have the [[Definition:Real Number Line|real number line]] $\R$ is a [[Definition:Vector Space|vector space]] over $\R$.
That the [[Axiom:Vector Space Norm Axioms|norm axioms]] are satisfied by the [[Definition:Absolute Value|absolute value function]] is proven in [[Absolute ... | Real Number Line is Banach Space | https://proofwiki.org/wiki/Real_Number_Line_is_Banach_Space | https://proofwiki.org/wiki/Real_Number_Line_is_Banach_Space | [
"Banach Spaces",
"Real Analysis"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Absolute Value",
"Definition:Banach Space"
] | [
"Real Numbers form Vector Space",
"Definition:Real Number/Real Number Line",
"Definition:Vector Space",
"Axiom:Vector Space Norm Axioms",
"Definition:Absolute Value",
"Absolute Value is Norm",
"Real Number Line is Complete Metric Space",
"Category:Banach Spaces",
"Category:Real Analysis"
] |
proofwiki-17832 | Convergent Subsequence of Cauchy Sequence/Normed Vector Space | Let $\struct {X, \norm {\,\cdot\,} }$ be a normed vector space.
Let $\sequence{x_n}_{n \mathop \in \N}$ be a Cauchy sequence in $\struct {X, \norm {\,\cdot\,} }$.
Let $x \in X$.
Then $\sequence {x_n}$ converges to $x$ {{iff}} $\sequence {x_n}$ has a subsequence that converges to $x$. | === Necessary Condition ===
Suppose is a Cauchy sequence$\sequence {x_n}$ which converges to $x$.
By Limit of Subsequence equals Limit of Sequence, $\sequence{x_n}$ has a subsequence of itself that converges to $x$.
{{qed|lemma}} | Let $\struct {X, \norm {\,\cdot\,} }$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $\sequence{x_n}_{n \mathop \in \N}$ be a [[Definition:Cauchy Sequence in Normed Vector Space|Cauchy sequence]] in $\struct {X, \norm {\,\cdot\,} }$.
Let $x \in X$.
Then $\sequence {x_n}$ [[Definition:Convergent S... | === Necessary Condition ===
Suppose is a [[Definition:Cauchy Sequence in Normed Vector Space|Cauchy sequence]]$\sequence {x_n}$ which [[Definition:Convergent Sequence in Normed Vector Space|converges]] to $x$.
By [[Limit of Subsequence equals Limit of Sequence/Normed Vector Space|Limit of Subsequence equals Limit of ... | Convergent Subsequence of Cauchy Sequence/Normed Vector Space | https://proofwiki.org/wiki/Convergent_Subsequence_of_Cauchy_Sequence/Normed_Vector_Space | https://proofwiki.org/wiki/Convergent_Subsequence_of_Cauchy_Sequence/Normed_Vector_Space | [
"Cauchy Sequences",
"Convergent Sequences (Normed Vector Spaces)"
] | [
"Definition:Normed Vector Space",
"Definition:Cauchy Sequence/Normed Vector Space",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Subsequence",
"Definition:Convergent Sequence/Normed Vector Space"
] | [
"Definition:Cauchy Sequence/Normed Vector Space",
"Definition:Convergent Sequence/Normed Vector Space",
"Limit of Subsequence equals Limit of Sequence/Normed Vector Space",
"Definition:Subsequence",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Subsequence",
"Definition:Convergent Se... |
proofwiki-17833 | Combination Theorem for Continuous Functions/Complex/Sum Rule | :$f + g$ is continuous on $S$. | We have:
{{begin-eqn}}
{{eqn | q = \forall c \in S
| l = \lim_{x \mathop \to c} \map f z
| r = \map f c
| c = {{Defof|Continuous Complex Function}}
}}
{{eqn | q = \forall c \in S
| l = \lim_{x \mathop \to c} \map g z
| r = \map g c
| c = {{Defof|Continuous Complex Function}}
}}
{{end... | :$f + g$ is [[Definition:Continuous Complex Function|continuous]] on $S$. | We have:
{{begin-eqn}}
{{eqn | q = \forall c \in S
| l = \lim_{x \mathop \to c} \map f z
| r = \map f c
| c = {{Defof|Continuous Complex Function}}
}}
{{eqn | q = \forall c \in S
| l = \lim_{x \mathop \to c} \map g z
| r = \map g c
| c = {{Defof|Continuous Complex Function}}
}}
{{end... | Combination Theorem for Continuous Functions/Complex/Sum Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Functions/Complex/Sum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Functions/Complex/Sum_Rule | [
"Combination Theorems for Continuous Functions"
] | [
"Definition:Continuous Complex Function"
] | [
"Definition:Limit of Complex Function",
"Combination Theorem for Limits of Functions/Complex/Sum Rule",
"Definition:Continuous Complex Function",
"Definition:Continuous Complex Function",
"Category:Combination Theorems for Continuous Functions"
] |
proofwiki-17834 | Combination Theorem for Continuous Functions/Complex/Multiple Rule | :$\lambda f$ is continuous on $S$. | By definition of continuous, we have that
:$\forall c \in S: \ds \lim_{z \mathop \to c} \map f z = \map f c$
Let $f$ tend to the following limit:
:$\ds \lim_{z \mathop \to c} \map f z = l$
From the Multiple Rule for Limits of Complex Functions, we have that:
:$\ds \lim_{z \mathop \to c} \paren {\lambda \map f z} = \lam... | :$\lambda f$ is [[Definition:Continuous Complex Function|continuous]] on $S$. | By definition of [[Definition:Continuous Complex Function|continuous]], we have that
:$\forall c \in S: \ds \lim_{z \mathop \to c} \map f z = \map f c$
Let $f$ tend to the following [[Definition:Limit of Complex Function|limit]]:
:$\ds \lim_{z \mathop \to c} \map f z = l$
From the [[Multiple Rule for Limits of Comp... | Combination Theorem for Continuous Functions/Complex/Multiple Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Functions/Complex/Multiple_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Functions/Complex/Multiple_Rule | [
"Combination Theorems for Continuous Functions"
] | [
"Definition:Continuous Complex Function"
] | [
"Definition:Continuous Complex Function",
"Definition:Limit of Complex Function",
"Combination Theorem for Limits of Functions/Complex/Multiple Rule",
"Definition:Continuous Complex Function",
"Definition:Continuous Complex Function",
"Category:Combination Theorems for Continuous Functions"
] |
proofwiki-17835 | Combination Theorem for Continuous Functions/Complex/Combined Sum Rule | :$\lambda f + \mu g$ is continuous on $S$. | By definition of continuous, we have that
:$\forall c \in S: \ds \lim_{z \mathop \to c} \map f z = \map f c$
:$\forall c \in S: \ds \lim_{z \mathop \to c} \map g z = \map g c$
Let $f$ and $g$ tend to the following limits:
:$\ds \lim_{z \mathop \to c} \map f x = l$
:$\ds \lim_{z \mathop \to c} \map g x = m$
From the Com... | :$\lambda f + \mu g$ is [[Definition:Continuous Complex Function|continuous]] on $S$. | By definition of [[Definition:Continuous Complex Function|continuous]], we have that
:$\forall c \in S: \ds \lim_{z \mathop \to c} \map f z = \map f c$
:$\forall c \in S: \ds \lim_{z \mathop \to c} \map g z = \map g c$
Let $f$ and $g$ tend to the following [[Definition:Limit of Complex Function|limits]]:
:$\ds \lim_{... | Combination Theorem for Continuous Functions/Complex/Combined Sum Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Functions/Complex/Combined_Sum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Functions/Complex/Combined_Sum_Rule | [
"Combination Theorems for Continuous Functions"
] | [
"Definition:Continuous Complex Function"
] | [
"Definition:Continuous Complex Function",
"Definition:Limit of Complex Function",
"Combination Theorem for Limits of Functions/Complex/Combined Sum Rule",
"Definition:Continuous Complex Function",
"Definition:Continuous Complex Function",
"Category:Combination Theorems for Continuous Functions"
] |
proofwiki-17836 | Combination Theorem for Continuous Functions/Complex/Product Rule | :$f g$ is continuous on $S$ | By definition of continuous:
:$\forall c \in S: \ds \lim_{z \mathop \to c} \map f z = \map f c$
:$\forall c \in S: \ds \lim_{z \mathop \to c} \map g z = \map g c$
Let $f$ and $g$ tend to the following limits:
:$\ds \lim_{z \mathop \to c} \map f z = l$
:$\ds \lim_{z \mathop \to c} \map g z = m$
From the Product Rule for... | :$f g$ is [[Definition:Continuous Complex Function|continuous]] on $S$ | By definition of [[Definition:Continuous Complex Function|continuous]]:
:$\forall c \in S: \ds \lim_{z \mathop \to c} \map f z = \map f c$
:$\forall c \in S: \ds \lim_{z \mathop \to c} \map g z = \map g c$
Let $f$ and $g$ tend to the following [[Definition:Limit of Complex Function|limits]]:
:$\ds \lim_{z \mathop \to... | Combination Theorem for Continuous Functions/Complex/Product Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Functions/Complex/Product_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Functions/Complex/Product_Rule | [
"Combination Theorems for Continuous Functions"
] | [
"Definition:Continuous Complex Function"
] | [
"Definition:Continuous Complex Function",
"Definition:Limit of Complex Function",
"Combination Theorem for Limits of Functions/Complex/Product Rule",
"Definition:Continuous Complex Function",
"Definition:Continuous Complex Function",
"Category:Combination Theorems for Continuous Functions"
] |
proofwiki-17837 | Combination Theorem for Continuous Functions/Complex/Quotient Rule | :$\dfrac f g$ is continuous on $S \setminus \set {z \in S: \map g z = 0}$
that is, on all the points $z$ of $S$ where $\map g z \ne 0$. | By definition of continuous:
:$\forall c \in S: \ds \lim_{z \mathop \to c} \map f z = \map f c$
:$\forall c \in S: \ds \lim_{z \mathop \to c} \map g z = \map g c$
Let $f$ and $g$ tend to the following limits:
:$\ds \lim_{z \mathop \to c} \map f z = l$
:$\ds \lim_{z \mathop \to c} \map g z = m$
From the Quotient Rule fo... | :$\dfrac f g$ is [[Definition:Continuous Complex Function|continuous]] on $S \setminus \set {z \in S: \map g z = 0}$
that is, on all the points $z$ of $S$ where $\map g z \ne 0$. | By definition of [[Definition:Continuous Complex Function|continuous]]:
:$\forall c \in S: \ds \lim_{z \mathop \to c} \map f z = \map f c$
:$\forall c \in S: \ds \lim_{z \mathop \to c} \map g z = \map g c$
Let $f$ and $g$ tend to the following [[Definition:Limit of Complex Function|limits]]:
:$\ds \lim_{z \mathop \to... | Combination Theorem for Continuous Functions/Complex/Quotient Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Functions/Complex/Quotient_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Functions/Complex/Quotient_Rule | [
"Combination Theorems for Continuous Functions"
] | [
"Definition:Continuous Complex Function"
] | [
"Definition:Continuous Complex Function",
"Definition:Limit of Complex Function",
"Combination Theorem for Limits of Functions/Complex/Quotient Rule",
"Definition:Continuous Complex Function",
"Definition:Continuous Complex Function",
"Category:Combination Theorems for Continuous Functions"
] |
proofwiki-17838 | Limit of Function by Convergent Sequences/Complex Plane | Let $f$ be a complex function defined on an open subset $S \subseteq \C$, except possibly at the point $c \in S$.
Then $\ds \lim_{x \mathop \to c} \map f z = l$ {{iff}}:
:for each sequence $\sequence {z_n}$ of points of $S$ such that $\forall n \in \N_{>0}: z_n \ne c$ and $\ds \lim_{n \mathop \to \infty} z_n = c$
it is... | === Necessary Condition ===
Let $\ds \lim_{z \mathop \to c} \map f z = l$.
Let $\epsilon \in \R_{>0}$.
Then by definition of the limit of a complex function:
:$\exists \delta \in \R_{>0}: \cmod {\map f z - l} < \epsilon$
provided $0 < \cmod {z - c} < \delta$.
Now suppose that $\sequence {x_n}$ is a sequence of elements... | Let $f$ be a [[Definition:Complex Function|complex function]] defined on an [[Definition:Open Set (Complex Analysis)|open subset]] $S \subseteq \C$, except possibly at the point $c \in S$.
Then $\ds \lim_{x \mathop \to c} \map f z = l$ {{iff}}:
:for each [[Definition:Real Sequence|sequence]] $\sequence {z_n}$ of poin... | === Necessary Condition ===
Let $\ds \lim_{z \mathop \to c} \map f z = l$.
Let $\epsilon \in \R_{>0}$.
Then by definition of the [[Definition:Limit of Complex Function|limit of a complex function]]:
:$\exists \delta \in \R_{>0}: \cmod {\map f z - l} < \epsilon$
provided $0 < \cmod {z - c} < \delta$.
Now suppose tha... | Limit of Function by Convergent Sequences/Complex Plane | https://proofwiki.org/wiki/Limit_of_Function_by_Convergent_Sequences/Complex_Plane | https://proofwiki.org/wiki/Limit_of_Function_by_Convergent_Sequences/Complex_Plane | [
"Limits of Complex Functions",
"Limits of Sequences"
] | [
"Definition:Complex Function",
"Definition:Open Set/Complex Analysis",
"Definition:Real Sequence"
] | [
"Definition:Limit of Complex Function",
"Definition:Complex Sequence",
"Definition:Element",
"Definition:Limit of Complex Function",
"Definition:Complex Sequence",
"Definition:Element",
"Definition:Complex Sequence",
"Definition:Element"
] |
proofwiki-17839 | Combination Theorem for Limits of Functions/Complex/Sum Rule | :$\ds \lim_{z \mathop \to c} \paren {\map f z + \map g z} = l + m$ | Let $\sequence {z_n}$ be any sequence of elements of $S$ such that:
:$\forall n \in \N_{>0}: z_n \ne c$
:$\ds \lim_{n \mathop \to \infty} z_n = c$
By Limit of Complex Function by Convergent Sequences:
:$\ds \lim_{n \mathop \to \infty} \map f {z_n} = l$
:$\ds \lim_{n \mathop \to \infty} \map g {z_n} = m$
By the Sum Rule... | :$\ds \lim_{z \mathop \to c} \paren {\map f z + \map g z} = l + m$ | Let $\sequence {z_n}$ be any [[Definition:Sequence|sequence]] of [[Definition:Element|elements]] of $S$ such that:
:$\forall n \in \N_{>0}: z_n \ne c$
:$\ds \lim_{n \mathop \to \infty} z_n = c$
By [[Limit of Complex Function by Convergent Sequences]]:
:$\ds \lim_{n \mathop \to \infty} \map f {z_n} = l$
:$\ds \lim_{n ... | Combination Theorem for Limits of Functions/Complex/Sum Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Limits_of_Functions/Complex/Sum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Limits_of_Functions/Complex/Sum_Rule | [
"Combination Theorems for Limits of Functions"
] | [] | [
"Definition:Sequence",
"Definition:Element",
"Limit of Function by Convergent Sequences/Complex Plane",
"Combination Theorem for Sequences/Complex/Sum Rule",
"Limit of Function by Convergent Sequences/Complex Plane",
"Category:Combination Theorems for Limits of Functions"
] |
proofwiki-17840 | Convergent Subsequence of Cauchy Sequence/Metric Space | Let $\struct {A, d}$ be a metric space.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a Cauchy sequence in $A$.
Let $x \in A$.
Then $\sequence {x_n}$ converges to $x$ {{iff}} it has a subsequence that converges to $x$. | === Necessary Condition ===
If $\sequence {x_n}$ converges to $x$, it trivially follows that $\sequence {x_n}$ is a subsequence of itself that converges to $x$.
{{qed|lemma}} | Let $\struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Cauchy Sequence in Metric Space|Cauchy sequence]] in $A$.
Let $x \in A$.
Then $\sequence {x_n}$ [[Definition:Convergent Sequence in Metric Space|converges]] to $x$ {{iff}} it has a [[Defin... | === Necessary Condition ===
If $\sequence {x_n}$ [[Definition:Convergent Sequence in Metric Space|converges]] to $x$, it trivially follows that $\sequence {x_n}$ is a [[Definition:Subsequence|subsequence]] of itself that [[Definition:Convergent Sequence (Metric Space)|converges]] to $x$.
{{qed|lemma}} | Convergent Subsequence of Cauchy Sequence/Metric Space | https://proofwiki.org/wiki/Convergent_Subsequence_of_Cauchy_Sequence/Metric_Space | https://proofwiki.org/wiki/Convergent_Subsequence_of_Cauchy_Sequence/Metric_Space | [
"Cauchy Sequences"
] | [
"Definition:Metric Space",
"Definition:Cauchy Sequence/Metric Space",
"Definition:Convergent Sequence/Metric Space",
"Definition:Subsequence",
"Definition:Convergent Sequence/Metric Space"
] | [
"Definition:Convergent Sequence/Metric Space",
"Definition:Subsequence",
"Definition:Convergent Sequence/Metric Space",
"Definition:Subsequence",
"Definition:Convergent Sequence/Metric Space",
"Definition:Convergent Sequence/Metric Space",
"Definition:Convergent Sequence/Metric Space"
] |
proofwiki-17841 | Combination Theorem for Limits of Functions/Complex/Multiple Rule | :$\ds \lim_{z \mathop \to c} \lambda \map f z = \lambda l$ | Let $\sequence {z_n}$ be any sequence of elements of $S$ such that:
:$\forall n \in \N_{>0}: z_n \ne c$
:$\ds \lim_{n \mathop \to \infty} z_n = c$
By Limit of Complex Function by Convergent Sequences:
:$\ds \lim_{n \mathop \to \infty} \map f {z_n} = l$
By the Multiple Rule for Complex Sequences:
:$\ds \lim_{n \mathop \... | :$\ds \lim_{z \mathop \to c} \lambda \map f z = \lambda l$ | Let $\sequence {z_n}$ be any [[Definition:Sequence|sequence]] of [[Definition:Element|elements]] of $S$ such that:
:$\forall n \in \N_{>0}: z_n \ne c$
:$\ds \lim_{n \mathop \to \infty} z_n = c$
By [[Limit of Complex Function by Convergent Sequences]]:
:$\ds \lim_{n \mathop \to \infty} \map f {z_n} = l$
By the [[Mul... | Combination Theorem for Limits of Functions/Complex/Multiple Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Limits_of_Functions/Complex/Multiple_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Limits_of_Functions/Complex/Multiple_Rule | [
"Combination Theorems for Limits of Functions"
] | [] | [
"Definition:Sequence",
"Definition:Element",
"Limit of Function by Convergent Sequences/Complex Plane",
"Combination Theorem for Sequences/Complex/Multiple Rule",
"Limit of Function by Convergent Sequences/Complex Plane",
"Category:Combination Theorems for Limits of Functions"
] |
proofwiki-17842 | Combination Theorem for Limits of Functions/Complex/Combined Sum Rule | :$\ds \lim_{z \mathop \to c} \paren {\lambda \map f z + \mu \map g z} = \lambda l + \mu m$ | Let $\sequence {z_n}$ be any sequence of elements of $S$ such that:
:$\forall n \in \N_{>0}: z_n \ne c$
:$\ds \lim_{n \mathop \to \infty} z_n = c$
By Limit of Complex Function by Convergent Sequences:
:$\ds \lim_{n \mathop \to \infty} \map f {z_n} = l$
:$\ds \lim_{n \mathop \to \infty} \map g {z_n} = m$
By the Combined... | :$\ds \lim_{z \mathop \to c} \paren {\lambda \map f z + \mu \map g z} = \lambda l + \mu m$ | Let $\sequence {z_n}$ be any [[Definition:Sequence|sequence]] of [[Definition:Element|elements]] of $S$ such that:
:$\forall n \in \N_{>0}: z_n \ne c$
:$\ds \lim_{n \mathop \to \infty} z_n = c$
By [[Limit of Complex Function by Convergent Sequences]]:
:$\ds \lim_{n \mathop \to \infty} \map f {z_n} = l$
:$\ds \lim_{n ... | Combination Theorem for Limits of Functions/Complex/Combined Sum Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Limits_of_Functions/Complex/Combined_Sum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Limits_of_Functions/Complex/Combined_Sum_Rule | [
"Combination Theorems for Limits of Functions"
] | [] | [
"Definition:Sequence",
"Definition:Element",
"Limit of Function by Convergent Sequences/Complex Plane",
"Combination Theorem for Sequences/Complex/Combined Sum Rule",
"Limit of Function by Convergent Sequences/Complex Plane",
"Category:Combination Theorems for Limits of Functions"
] |
proofwiki-17843 | Combination Theorem for Limits of Functions/Complex/Product Rule | :$\ds \lim_{z \mathop \to c} \ \paren {\map f z \map g z} = l m$ | Let $\sequence {z_n}$ be a sequence of elements of $S$ such that:
:$\forall n \in \N: z_n \ne c$
:$\ds \lim_{n \mathop \to \infty} z_n = c$
By Limit of Complex Function by Convergent Sequences:
:$\ds \lim_{n \mathop \to \infty} \map f {z_n} = l$
:$\ds \lim_{n \mathop \to \infty} \map g {z_n} = m$
By the Product Rule fo... | :$\ds \lim_{z \mathop \to c} \ \paren {\map f z \map g z} = l m$ | Let $\sequence {z_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Element|elements]] of $S$ such that:
:$\forall n \in \N: z_n \ne c$
:$\ds \lim_{n \mathop \to \infty} z_n = c$
By [[Limit of Complex Function by Convergent Sequences]]:
:$\ds \lim_{n \mathop \to \infty} \map f {z_n} = l$
:$\ds \lim_{n \mathop... | Combination Theorem for Limits of Functions/Complex/Product Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Limits_of_Functions/Complex/Product_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Limits_of_Functions/Complex/Product_Rule | [
"Combination Theorems for Limits of Functions"
] | [] | [
"Definition:Sequence",
"Definition:Element",
"Limit of Function by Convergent Sequences/Complex Plane",
"Combination Theorem for Sequences/Complex/Product Rule",
"Limit of Function by Convergent Sequences/Complex Plane",
"Category:Combination Theorems for Limits of Functions"
] |
proofwiki-17844 | Combination Theorem for Limits of Functions/Complex/Quotient Rule | :$\ds \lim_{z \mathop \to c} \frac {\map f z} {\map g z} = \frac l m$
provided that $m \ne 0$. | Let $\sequence {z_n}$ be any sequence of elements of $S$ such that:
:$\forall n \in \N_{>0}: z_n \ne c$
:$\ds \lim_{n \mathop \to \infty} z_n = c$
By Limit of Real Function by Convergent Sequences:
:$\ds \lim_{n \mathop \to \infty} \map f {z_n} = l$
:$\ds \lim_{n \mathop \to \infty} \map g {z_n} = m$
By the Quotient Ru... | :$\ds \lim_{z \mathop \to c} \frac {\map f z} {\map g z} = \frac l m$
provided that $m \ne 0$. | Let $\sequence {z_n}$ be any [[Definition:Sequence|sequence]] of [[Definition:Element|elements]] of $S$ such that:
:$\forall n \in \N_{>0}: z_n \ne c$
:$\ds \lim_{n \mathop \to \infty} z_n = c$
By [[Limit of Real Function by Convergent Sequences]]:
:$\ds \lim_{n \mathop \to \infty} \map f {z_n} = l$
:$\ds \lim_{n \ma... | Combination Theorem for Limits of Functions/Complex/Quotient Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Limits_of_Functions/Complex/Quotient_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Limits_of_Functions/Complex/Quotient_Rule | [
"Combination Theorems for Limits of Functions"
] | [] | [
"Definition:Sequence",
"Definition:Element",
"Limit of Function by Convergent Sequences/Real Number Line",
"Combination Theorem for Sequences/Real/Quotient Rule",
"Limit of Function by Convergent Sequences/Real Number Line",
"Category:Combination Theorems for Limits of Functions"
] |
proofwiki-17845 | Subset of Bounded Above Set is Bounded Above | Let $A$ and $B$ be sets of real numbers such that $A \subseteq B$.
Let $B$ be bounded above.
Then $A$ is also bounded above. | Let $B$ be bounded above.
Then by definition $B$ has an upper bound $U$.
Hence:
:$\forall x \in B: x \le U$
But by definition of subset:
:$\forall x \in A: x \in B$
That is:
:$\forall x \in A: x \le U$
Hence, by definition, $A$ is bounded above by $U$.
{{qed}} | Let $A$ and $B$ be [[Definition:Set|sets]] of [[Definition:Real Number|real numbers]] such that $A \subseteq B$.
Let $B$ be [[Definition:Bounded Above Subset of Real Numbers|bounded above]].
Then $A$ is also [[Definition:Bounded Above Subset of Real Numbers|bounded above]]. | Let $B$ be [[Definition:Bounded Above Subset of Real Numbers|bounded above]].
Then by definition $B$ has an [[Definition:Upper Bound|upper bound]] $U$.
Hence:
:$\forall x \in B: x \le U$
But by definition of [[Definition:Subset|subset]]:
:$\forall x \in A: x \in B$
That is:
:$\forall x \in A: x \le U$
Hence, by de... | Subset of Bounded Above Set is Bounded Above | https://proofwiki.org/wiki/Subset_of_Bounded_Above_Set_is_Bounded_Above | https://proofwiki.org/wiki/Subset_of_Bounded_Above_Set_is_Bounded_Above | [
"Bounded Above Sets of Real Numbers"
] | [
"Definition:Set",
"Definition:Real Number",
"Definition:Bounded Above Set/Real Numbers",
"Definition:Bounded Above Set/Real Numbers"
] | [
"Definition:Bounded Above Set/Real Numbers",
"Definition:Upper Bound",
"Definition:Subset",
"Definition:Bounded Above Set/Real Numbers"
] |
proofwiki-17846 | Supremum of Subset of Bounded Above Set of Real Numbers | Let $A$ and $B$ be sets of real numbers such that $A \subseteq B$.
Let $B$ be bounded above.
Then:
:$\sup A \le \sup B$
where $\sup$ denotes the supremum. | Let $B$ be bounded above.
By the Continuum Property, $B$ admits a supremum.
By Subset of Bounded Above Set is Bounded Above, $A$ is also bounded above.
Hence also by the Continuum Property, $A$ also admits a supremum.
{{AimForCont}} $\sup A > \sup B$.
Then:
:$\exists y \in A: y > \sup B$
Thus by definition of supremum,... | Let $A$ and $B$ be [[Definition:Set|sets]] of [[Definition:Real Number|real numbers]] such that $A \subseteq B$.
Let $B$ be [[Definition:Bounded Above Subset of Real Numbers|bounded above]].
Then:
:$\sup A \le \sup B$
where $\sup$ denotes the [[Definition:Supremum of Subset of Real Numbers|supremum]]. | Let $B$ be [[Definition:Bounded Above Subset of Real Numbers|bounded above]].
By the [[Continuum Property]], $B$ admits a [[Definition:Supremum of Subset of Real Numbers|supremum]].
By [[Subset of Bounded Above Set is Bounded Above]], $A$ is also [[Definition:Bounded Above Subset of Real Numbers|bounded above]].
Hen... | Supremum of Subset of Bounded Above Set of Real Numbers | https://proofwiki.org/wiki/Supremum_of_Subset_of_Bounded_Above_Set_of_Real_Numbers | https://proofwiki.org/wiki/Supremum_of_Subset_of_Bounded_Above_Set_of_Real_Numbers | [
"Bounded Above Sets of Real Numbers",
"Suprema"
] | [
"Definition:Set",
"Definition:Real Number",
"Definition:Bounded Above Set/Real Numbers",
"Definition:Supremum of Set/Real Numbers"
] | [
"Definition:Bounded Above Set/Real Numbers",
"Continuum Property",
"Definition:Supremum of Set/Real Numbers",
"Subset of Bounded Above Set is Bounded Above",
"Definition:Bounded Above Set/Real Numbers",
"Continuum Property",
"Definition:Supremum of Set/Real Numbers",
"Definition:Supremum of Set/Real N... |
proofwiki-17847 | Union of Bounded Above Real Subsets is Bounded Above | Let $A$ and $B$ be sets of real numbers.
Let $A$ and $B$ be bounded above.
Then $A \cup B$ is also bounded above. | Let $A$ and $B$ both be bounded above.
Then by definition $A$ and $B$ both have an upper bound $U_A$ and $U_B$ respectively.
Suppose $U_A \le U_B$.
Then:
:$\forall a \in A: a \le U_B$
and also, by definition:
:$\forall b \in B: b \le U_B$
and so $U_B$ is an upper bound for $A$.
Otherwise, suppose $U_A > U_B$.
Then:
:$\... | Let $A$ and $B$ be [[Definition:Set|sets]] of [[Definition:Real Number|real numbers]].
Let $A$ and $B$ be [[Definition:Bounded Above Subset of Real Numbers|bounded above]].
Then $A \cup B$ is also [[Definition:Bounded Above Subset of Real Numbers|bounded above]]. | Let $A$ and $B$ both be [[Definition:Bounded Above Subset of Real Numbers|bounded above]].
Then by definition $A$ and $B$ both have an [[Definition:Upper Bound of Subset of Real Numbers|upper bound]] $U_A$ and $U_B$ respectively.
Suppose $U_A \le U_B$.
Then:
:$\forall a \in A: a \le U_B$
and also, by definition:
:$\... | Union of Bounded Above Real Subsets is Bounded Above | https://proofwiki.org/wiki/Union_of_Bounded_Above_Real_Subsets_is_Bounded_Above | https://proofwiki.org/wiki/Union_of_Bounded_Above_Real_Subsets_is_Bounded_Above | [
"Bounded Above Sets of Real Numbers",
"Set Union"
] | [
"Definition:Set",
"Definition:Real Number",
"Definition:Bounded Above Set/Real Numbers",
"Definition:Bounded Above Set/Real Numbers"
] | [
"Definition:Bounded Above Set/Real Numbers",
"Definition:Upper Bound of Set/Real Numbers",
"Definition:Upper Bound of Set/Real Numbers",
"Definition:Upper Bound of Set/Real Numbers",
"Definition:Bounded Above Set/Real Numbers"
] |
proofwiki-17848 | Supremum of Union of Bounded Above Sets of Real Numbers | Let $A$ and $B$ be sets of real numbers.
Let $A$ and $B$ both be bounded above.
Then:
:$\map \sup {A \cup B} = \max \set {\sup A, \sup B}$
where $\sup$ denotes the supremum. | Let $A$ and $B$ both be bounded above.
By the Continuum Property, $A$ and $B$ both admit a supremum.
Let $x \in A \cup B$.
Then either $x \le \sup A$ or $x \le \sup B$ by definition of supremum.
Hence:
:$x \le \max \set {\sup A, \sup B}$
and so $\max \set {\sup A, \sup B}$ is certainly an upper bound of $A \cup B$.
It ... | Let $A$ and $B$ be [[Definition:Set|sets]] of [[Definition:Real Number|real numbers]].
Let $A$ and $B$ both be [[Definition:Bounded Above Subset of Real Numbers|bounded above]].
Then:
:$\map \sup {A \cup B} = \max \set {\sup A, \sup B}$
where $\sup$ denotes the [[Definition:Supremum of Subset of Real Numbers|suprem... | Let $A$ and $B$ both be [[Definition:Bounded Above Subset of Real Numbers|bounded above]].
By the [[Continuum Property]], $A$ and $B$ both admit a [[Definition:Supremum of Subset of Real Numbers|supremum]].
Let $x \in A \cup B$.
Then either $x \le \sup A$ or $x \le \sup B$ by definition of [[Definition:Supremum of ... | Supremum of Union of Bounded Above Sets of Real Numbers | https://proofwiki.org/wiki/Supremum_of_Union_of_Bounded_Above_Sets_of_Real_Numbers | https://proofwiki.org/wiki/Supremum_of_Union_of_Bounded_Above_Sets_of_Real_Numbers | [
"Bounded Above Sets of Real Numbers",
"Suprema",
"Set Union"
] | [
"Definition:Set",
"Definition:Real Number",
"Definition:Bounded Above Set/Real Numbers",
"Definition:Supremum of Set/Real Numbers"
] | [
"Definition:Bounded Above Set/Real Numbers",
"Continuum Property",
"Definition:Supremum of Set/Real Numbers",
"Definition:Supremum of Set/Real Numbers",
"Definition:Upper Bound of Set/Real Numbers",
"Definition:Upper Bound of Set/Real Numbers",
"Definition:Supremum of Set/Real Numbers",
"Definition:Up... |
proofwiki-17849 | Subset of Bounded Below Set is Bounded Below | Let $A$ and $B$ be sets of real numbers such that $A \subseteq B$.
Let $B$ be bounded below.
Then $A$ is also bounded below. | Let $B$ be bounded below.
Then by definition $B$ has a lower bound $L$.
Hence:
:$\forall x \in B: x \ge L$
But by definition of subset:
:$\forall x \in A: x \in B$
That is:
:$\forall x \in A: x \ge L$
Hence, by definition, $A$ is bounded below by $L$.
{{qed}} | Let $A$ and $B$ be [[Definition:Set|sets]] of [[Definition:Real Number|real numbers]] such that $A \subseteq B$.
Let $B$ be [[Definition:Bounded Below Subset of Real Numbers|bounded below]].
Then $A$ is also [[Definition:Bounded Below Subset of Real Numbers|bounded below]]. | Let $B$ be [[Definition:Bounded Below Subset of Real Numbers|bounded below]].
Then by definition $B$ has a [[Definition:Lower Bound|lower bound]] $L$.
Hence:
:$\forall x \in B: x \ge L$
But by definition of [[Definition:Subset|subset]]:
:$\forall x \in A: x \in B$
That is:
:$\forall x \in A: x \ge L$
Hence, by def... | Subset of Bounded Below Set is Bounded Below | https://proofwiki.org/wiki/Subset_of_Bounded_Below_Set_is_Bounded_Below | https://proofwiki.org/wiki/Subset_of_Bounded_Below_Set_is_Bounded_Below | [
"Bounded Below Sets of Real Numbers"
] | [
"Definition:Set",
"Definition:Real Number",
"Definition:Bounded Below Set/Real Numbers",
"Definition:Bounded Below Set/Real Numbers"
] | [
"Definition:Bounded Below Set/Real Numbers",
"Definition:Lower Bound",
"Definition:Subset",
"Definition:Bounded Below Set/Real Numbers"
] |
proofwiki-17850 | Infimum of Subset of Bounded Below Set of Real Numbers | Let $A$ and $B$ be sets of real numbers such that $A \subseteq B$.
Let $B$ be bounded below.
Then:
:$\inf A \ge \inf B$
where $\inf$ denotes the infimum. | Let $B$ be bounded below.
By the Continuum Property, $B$ admits an infimum.
By Subset of Bounded Below Set is Bounded Below, $A$ is also bounded below.
Hence also by the Continuum Property, $A$ also admits a infimum.
{{AimForCont}} $\inf A < \inf B$.
Then:
:$\exists y \in A: y < \inf B$
Thus by definition of infimum, $... | Let $A$ and $B$ be [[Definition:Set|sets]] of [[Definition:Real Number|real numbers]] such that $A \subseteq B$.
Let $B$ be [[Definition:Bounded Below Subset of Real Numbers|bounded below]].
Then:
:$\inf A \ge \inf B$
where $\inf$ denotes the [[Definition:Infimum of Subset of Real Numbers|infimum]]. | Let $B$ be [[Definition:Bounded Below Subset of Real Numbers|bounded below]].
By the [[Continuum Property]], $B$ admits an [[Definition:Infimum of Subset of Real Numbers|infimum]].
By [[Subset of Bounded Below Set is Bounded Below]], $A$ is also [[Definition:Bounded Below Subset of Real Numbers|bounded below]].
Henc... | Infimum of Subset of Bounded Below Set of Real Numbers | https://proofwiki.org/wiki/Infimum_of_Subset_of_Bounded_Below_Set_of_Real_Numbers | https://proofwiki.org/wiki/Infimum_of_Subset_of_Bounded_Below_Set_of_Real_Numbers | [
"Bounded Below Sets of Real Numbers"
] | [
"Definition:Set",
"Definition:Real Number",
"Definition:Bounded Below Set/Real Numbers",
"Definition:Infimum of Set/Real Numbers"
] | [
"Definition:Bounded Below Set/Real Numbers",
"Continuum Property",
"Definition:Infimum of Set/Real Numbers",
"Subset of Bounded Below Set is Bounded Below",
"Definition:Bounded Below Set/Real Numbers",
"Continuum Property",
"Definition:Infimum of Set/Real Numbers",
"Definition:Infimum of Set/Real Numb... |
proofwiki-17851 | Infimum of Union of Bounded Below Sets of Real Numbers | Let $A$ and $B$ be sets of real numbers.
Let $A$ and $B$ both be bounded below.
Then:
:$\map \inf {A \cup B} = \min \set {\inf A, \inf B}$
where $\inf$ denotes the infimum. | Let $A$ and $B$ both be bounded below.
By the Continuum Property, $A$ and $B$ both admit an infimum.
Let $x \in A \cup B$.
Then either $x \ge \inf A$ or $x \ge \inf B$ by definition of infimum.
Hence:
:$x \ge \min \set {\inf A, \inf B}$
and so $\min \set {\inf A, \inf B}$ is certainly a lower bound of $A \cup B$.
It re... | Let $A$ and $B$ be [[Definition:Set|sets]] of [[Definition:Real Number|real numbers]].
Let $A$ and $B$ both be [[Definition:Bounded Below Subset of Real Numbers|bounded below]].
Then:
:$\map \inf {A \cup B} = \min \set {\inf A, \inf B}$
where $\inf$ denotes the [[Definition:Infimum of Subset of Real Numbers|infimum... | Let $A$ and $B$ both be [[Definition:Bounded Below Subset of Real Numbers|bounded below]].
By the [[Continuum Property]], $A$ and $B$ both admit an [[Definition:Infimum of Subset of Real Numbers|infimum]].
Let $x \in A \cup B$.
Then either $x \ge \inf A$ or $x \ge \inf B$ by definition of [[Definition:Infimum of Su... | Infimum of Union of Bounded Below Sets of Real Numbers | https://proofwiki.org/wiki/Infimum_of_Union_of_Bounded_Below_Sets_of_Real_Numbers | https://proofwiki.org/wiki/Infimum_of_Union_of_Bounded_Below_Sets_of_Real_Numbers | [
"Bounded Below Sets of Real Numbers",
"Infima",
"Set Union"
] | [
"Definition:Set",
"Definition:Real Number",
"Definition:Bounded Below Set/Real Numbers",
"Definition:Infimum of Set/Real Numbers"
] | [
"Definition:Bounded Below Set/Real Numbers",
"Continuum Property",
"Definition:Infimum of Set/Real Numbers",
"Definition:Infimum of Set/Real Numbers",
"Definition:Lower Bound of Set/Real Numbers",
"Definition:Lower Bound of Set/Real Numbers",
"Definition:Infimum of Set/Real Numbers",
"Definition:Lower... |
proofwiki-17852 | Union of Bounded Below Real Subsets is Bounded Below | Let $A$ and $B$ be sets of real numbers.
Let $A$ and $B$ be bounded below.
Then $A \cup B$ is also bounded below. | Let $A$ and $B$ both be bounded below.
Then by definition $A$ and $B$ both have a lower bound $L_A$ and $L_B$ respectively.
Suppose $L_A \ge L_B$.
Then:
:$\forall a \in A: a \ge L_B$
and also, by definition:
:$\forall b \in B: b \ge L_B$
and so $L_B$ is a lower bound for $A$.
Otherwise, suppose $L_A < L_B$.
Then:
:$\fo... | Let $A$ and $B$ be [[Definition:Set|sets]] of [[Definition:Real Number|real numbers]].
Let $A$ and $B$ be [[Definition:Bounded Below Subset of Real Numbers|bounded below]].
Then $A \cup B$ is also [[Definition:Bounded Below Subset of Real Numbers|bounded below]]. | Let $A$ and $B$ both be [[Definition:Bounded Below Subset of Real Numbers|bounded below]].
Then by definition $A$ and $B$ both have a [[Definition:Lower Bound of Subset of Real Numbers|lower bound]] $L_A$ and $L_B$ respectively.
Suppose $L_A \ge L_B$.
Then:
:$\forall a \in A: a \ge L_B$
and also, by definition:
:$\f... | Union of Bounded Below Real Subsets is Bounded Below | https://proofwiki.org/wiki/Union_of_Bounded_Below_Real_Subsets_is_Bounded_Below | https://proofwiki.org/wiki/Union_of_Bounded_Below_Real_Subsets_is_Bounded_Below | [
"Bounded Below Sets of Real Numbers",
"Set Union"
] | [
"Definition:Set",
"Definition:Real Number",
"Definition:Bounded Below Set/Real Numbers",
"Definition:Bounded Below Set/Real Numbers"
] | [
"Definition:Bounded Below Set/Real Numbers",
"Definition:Lower Bound of Set/Real Numbers",
"Definition:Lower Bound of Set/Real Numbers",
"Definition:Lower Bound of Set/Real Numbers",
"Definition:Bounded Below Set/Real Numbers"
] |
proofwiki-17853 | Condition for Rational Number to be Square of Rational Number | Let $m$ and $n$ be (strictly) positive integers which are coprime.
Then:
:$\dfrac m n$ is the square of a rational number
{{iff}}:
:both $m$ and $n$ are square numbers. | Let $m$ and $n$ be (strictly) positive integers which are coprime. | Let $m$ and $n$ be [[Definition:Strictly Positive Integer|(strictly) positive integers]] which are [[Definition:Coprime Integers|coprime]].
Then:
:$\dfrac m n$ is the [[Definition:Square Function|square]] of a [[Definition:Rational Number|rational number]]
{{iff}}:
:both $m$ and $n$ are [[Definition:Square Number|squa... | Let $m$ and $n$ be [[Definition:Strictly Positive Integer|(strictly) positive integers]] which are [[Definition:Coprime Integers|coprime]]. | Condition for Rational Number to be Square of Rational Number | https://proofwiki.org/wiki/Condition_for_Rational_Number_to_be_Square_of_Rational_Number | https://proofwiki.org/wiki/Condition_for_Rational_Number_to_be_Square_of_Rational_Number | [
"Square Roots",
"Rational Numbers",
"Square Numbers"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Coprime/Integers",
"Definition:Square/Function",
"Definition:Rational Number",
"Definition:Square Number"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Coprime/Integers",
"Definition:Coprime/Integers"
] |
proofwiki-17854 | Greatest Lower Bound Property | Let $S \subset \R$ be a non-empty subset of the set of real numbers such that $S$ is bounded below.
Then $S$ admits an infimum in $\R$.
This is known as the '''greatest lower bound property''' of the real numbers. | Let $T = \set {x \in \R: -x \in S}$.
By Set of Real Numbers is Bounded Below iff Set of Negatives is Bounded Above:
:$T$ is bounded above.
Thus by the Least Upper Bound Property, $T$ admits a supremum in $\R$.
From Negative of Supremum is Infimum of Negatives:
:$\ds -\sup_{x \mathop \in T} x = \map {\inf_{x \mathop \in... | Let $S \subset \R$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of the [[Definition:Real Number|set of real numbers]] such that $S$ is [[Definition:Bounded Below Subset of Real Numbers|bounded below]].
Then $S$ [[Definition:Infimum of Subset of Real Numbers|admits an infimum]] in $\R$.
Thi... | Let $T = \set {x \in \R: -x \in S}$.
By [[Set of Real Numbers is Bounded Below iff Set of Negatives is Bounded Above]]:
:$T$ is [[Definition:Bounded Above Subset of Real Numbers|bounded above]].
Thus by the [[Least Upper Bound Property]], $T$ [[Definition:Supremum of Subset of Real Numbers|admits a supremum]] in $\R$... | Greatest Lower Bound Property/Proof 1 | https://proofwiki.org/wiki/Greatest_Lower_Bound_Property | https://proofwiki.org/wiki/Greatest_Lower_Bound_Property/Proof_1 | [
"Continuum Property",
"Named Theorems"
] | [
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Real Number",
"Definition:Bounded Below Set/Real Numbers",
"Definition:Infimum of Set/Real Numbers",
"Greatest Lower Bound Property",
"Definition:Real Number"
] | [
"Set of Real Numbers is Bounded Below iff Set of Negatives is Bounded Above",
"Definition:Bounded Above Set/Real Numbers",
"Least Upper Bound Property",
"Definition:Supremum of Set/Real Numbers",
"Negative of Supremum is Infimum of Negatives",
"Definition:Infimum of Set/Real Numbers"
] |
proofwiki-17855 | Greatest Lower Bound Property | Let $S \subset \R$ be a non-empty subset of the set of real numbers such that $S$ is bounded below.
Then $S$ admits an infimum in $\R$.
This is known as the '''greatest lower bound property''' of the real numbers. | Let $T$ be the set of lower bounds of $S$:
:$T = \set {x \in \R: x \le \forall y \in S}$
Since $S$ is bounded below, $T$ is non-empty.
Now, every $x \in T$ and $y \in S$ satisfy $x \leq y$.
That is, $T$ is bounded above by every element of $S$.
By the Least Upper Bound Property, $T$ admits a supremum in $\R$.
Let $B = ... | Let $S \subset \R$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of the [[Definition:Real Number|set of real numbers]] such that $S$ is [[Definition:Bounded Below Subset of Real Numbers|bounded below]].
Then $S$ [[Definition:Infimum of Subset of Real Numbers|admits an infimum]] in $\R$.
Thi... | Let $T$ be the set of [[Definition:Lower Bound of Subset of Real Numbers|lower bounds]] of $S$:
:$T = \set {x \in \R: x \le \forall y \in S}$
Since $S$ is [[Definition:Bounded Below Subset of Real Numbers|bounded below]], $T$ is [[Definition:Non-Empty|non-empty]].
Now, every $x \in T$ and $y \in S$ satisfy $x \leq ... | Greatest Lower Bound Property/Proof 2 | https://proofwiki.org/wiki/Greatest_Lower_Bound_Property | https://proofwiki.org/wiki/Greatest_Lower_Bound_Property/Proof_2 | [
"Continuum Property",
"Named Theorems"
] | [
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Real Number",
"Definition:Bounded Below Set/Real Numbers",
"Definition:Infimum of Set/Real Numbers",
"Greatest Lower Bound Property",
"Definition:Real Number"
] | [
"Definition:Lower Bound of Set/Real Numbers",
"Definition:Bounded Below Set/Real Numbers",
"Definition:Non-Empty",
"Definition:Bounded Above Set/Real Numbers",
"Least Upper Bound Property",
"Definition:Supremum of Set/Real Numbers",
"Definition:Element",
"Definition:Upper Bound of Set/Real Numbers",
... |
proofwiki-17856 | Set of Real Numbers is Bounded Below iff Set of Negatives is Bounded Above | Let $S$ be a subset of the real numbers $\R$.
Let $T = \set {x \in \R: -x \in S}$ be the set of negatives of the elements of $S$.
Then:
:$S$ is bounded below
{{iff}}:
:$T$ is bounded above. | === Sufficient Condition ===
Let $S$ be bounded below.
Then $S$ has a lower bound.
Let $B$ be a lower bound for $S$.
From Negative of Lower Bound of Set of Real Numbers is Upper Bound of Negatives:
:$B$ is a lower bound for $S$
{{iff}}
:$-B$ is an upper bound for $T$.
It follows that $T$ is bounded above.
{{qed|lemma}} | Let $S$ be a [[Definition:Subset|subset]] of the [[Definition:Real Number|real numbers]] $\R$.
Let $T = \set {x \in \R: -x \in S}$ be the [[Definition:Set|set]] of [[Definition:Negative of Real Number|negatives]] of the [[Definition:Element|elements]] of $S$.
Then:
:$S$ is [[Definition:Bounded Below Subset of Real N... | === Sufficient Condition ===
Let $S$ be [[Definition:Bounded Below Subset of Real Numbers|bounded below]].
Then $S$ has a [[Definition:Lower Bound of Subset of Real Numbers|lower bound]].
Let $B$ be a [[Definition:Lower Bound of Subset of Real Numbers|lower bound]] for $S$.
From [[Negative of Lower Bound of Set of... | Set of Real Numbers is Bounded Below iff Set of Negatives is Bounded Above | https://proofwiki.org/wiki/Set_of_Real_Numbers_is_Bounded_Below_iff_Set_of_Negatives_is_Bounded_Above | https://proofwiki.org/wiki/Set_of_Real_Numbers_is_Bounded_Below_iff_Set_of_Negatives_is_Bounded_Above | [
"Real Analysis"
] | [
"Definition:Subset",
"Definition:Real Number",
"Definition:Set",
"Definition:Negative of Real Number",
"Definition:Element",
"Definition:Bounded Below Set/Real Numbers",
"Definition:Bounded Above Set/Real Numbers"
] | [
"Definition:Bounded Below Set/Real Numbers",
"Definition:Lower Bound of Set/Real Numbers",
"Definition:Lower Bound of Set/Real Numbers",
"Negative of Lower Bound of Set of Real Numbers is Upper Bound of Negatives",
"Definition:Lower Bound of Set/Real Numbers",
"Definition:Upper Bound of Set/Real Numbers",... |
proofwiki-17857 | Negative of Lower Bound of Set of Real Numbers is Upper Bound of Negatives | Let $S$ be a subset of the real numbers $\R$.
Let $T = \set {x \in \R: -x \in S}$ be the set of negatives of the elements of $S$.
Then:
:$B$ is a lower bound of $S$
{{iff}}:
:$-B$ is an upper bound of $T$. | Let $B$ be a lower bound of $S$.
That is:
:$\forall x \in S: x \ge B$
Let $x \in T$ be arbitrary.
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = T
}}
{{eqn | ll= \leadsto
| l = -x
| o = \in
| r = S
| c =
}}
{{eqn | ll= \leadsto
| l = -x
| o = \ge
| r = B
| c = a... | Let $S$ be a [[Definition:Subset|subset]] of the [[Definition:Real Number|real numbers]] $\R$.
Let $T = \set {x \in \R: -x \in S}$ be the [[Definition:Set|set]] of [[Definition:Negative of Real Number|negatives]] of the [[Definition:Element|elements]] of $S$.
Then:
:$B$ is a [[Definition:Lower Bound of Subset of Rea... | Let $B$ be a [[Definition:Lower Bound of Subset of Real Numbers|lower bound]] of $S$.
That is:
:$\forall x \in S: x \ge B$
Let $x \in T$ be arbitrary.
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = T
}}
{{eqn | ll= \leadsto
| l = -x
| o = \in
| r = S
| c =
}}
{{eqn | ll= \leadsto
... | Negative of Lower Bound of Set of Real Numbers is Upper Bound of Negatives | https://proofwiki.org/wiki/Negative_of_Lower_Bound_of_Set_of_Real_Numbers_is_Upper_Bound_of_Negatives | https://proofwiki.org/wiki/Negative_of_Lower_Bound_of_Set_of_Real_Numbers_is_Upper_Bound_of_Negatives | [
"Real Analysis"
] | [
"Definition:Subset",
"Definition:Real Number",
"Definition:Set",
"Definition:Negative of Real Number",
"Definition:Element",
"Definition:Lower Bound of Set/Real Numbers",
"Definition:Upper Bound of Set/Real Numbers"
] | [
"Definition:Lower Bound of Set/Real Numbers",
"Definition:Lower Bound of Set/Real Numbers",
"Ordering of Real Numbers is Reversed by Negation",
"Definition:Upper Bound of Set/Real Numbers",
"Definition:Upper Bound of Set/Real Numbers",
"Definition:Upper Bound of Set/Real Numbers",
"Ordering of Real Numb... |
proofwiki-17858 | Ordering of Real Numbers is Reversed by Negation | Let $x$ and $y$ in $\R$ be real numbers such that:
:$x \le y$
Then:
:$-y \le -x$
where $-y$ and $-x$ are the negatives of $y$ and $x$ respectively. | By definition of ordering:
:$x \le y$
{{iff}}:
:$x < y \text { or } x = y$
From Order of Real Numbers is Dual of Order of their Negatives:
:$x < y \iff \paren {-x} > \paren {-y}$
Hence the result.
{{qed}}
Category:Real Numbers
trgpq86ye6amupvisdvcnjcweln53xy | Let $x$ and $y$ in $\R$ be [[Definition:Real Number|real numbers]] such that:
:$x \le y$
Then:
:$-y \le -x$
where $-y$ and $-x$ are the [[Definition:Negative of Real Number|negatives]] of $y$ and $x$ respectively. | By definition of [[Definition:Ordering|ordering]]:
:$x \le y$
{{iff}}:
:$x < y \text { or } x = y$
From [[Order of Real Numbers is Dual of Order of their Negatives]]:
:$x < y \iff \paren {-x} > \paren {-y}$
Hence the result.
{{qed}}
[[Category:Real Numbers]]
trgpq86ye6amupvisdvcnjcweln53xy | Ordering of Real Numbers is Reversed by Negation | https://proofwiki.org/wiki/Ordering_of_Real_Numbers_is_Reversed_by_Negation | https://proofwiki.org/wiki/Ordering_of_Real_Numbers_is_Reversed_by_Negation | [
"Real Numbers"
] | [
"Definition:Real Number",
"Definition:Negative of Real Number"
] | [
"Definition:Ordering",
"Order of Real Numbers is Dual of Order of their Negatives",
"Category:Real Numbers"
] |
proofwiki-17859 | Negative of Upper Bound of Set of Real Numbers is Lower Bound of Negatives | Let $S$ be a subset of the real numbers $\R$.
Let $T = \set {x \in \R: -x \in S}$ be the set of negatives of the elements of $S$.
Then:
:$U$ is an upper bound of $S$
{{iff}}:
:$-U$ is a lower bound of $T$. | Let $V$ be the set defined as:
:$V = \set {x \in \R: -x \in T}$
From Negative of Lower Bound of Set of Real Numbers is Upper Bound of Negatives:
:$B$ is a lower bound of $T$
{{iff}}:
:$-B$ is an upper bound of $V$
Then we have:
{{begin-eqn}}
{{eqn | l = V
| r = \set {x \in \R: -x \in T}
| c =
}}
{{eqn | r ... | Let $S$ be a [[Definition:Subset|subset]] of the [[Definition:Real Number|real numbers]] $\R$.
Let $T = \set {x \in \R: -x \in S}$ be the [[Definition:Set|set]] of [[Definition:Negative of Real Number|negatives]] of the [[Definition:Element|elements]] of $S$.
Then:
:$U$ is an [[Definition:Upper Bound of Subset of Re... | Let $V$ be the [[Definition:Set|set]] defined as:
:$V = \set {x \in \R: -x \in T}$
From [[Negative of Lower Bound of Set of Real Numbers is Upper Bound of Negatives]]:
:$B$ is a [[Definition:Lower Bound of Subset of Real Numbers|lower bound]] of $T$
{{iff}}:
:$-B$ is an [[Definition:Upper Bound of Subset of Real Nu... | Negative of Upper Bound of Set of Real Numbers is Lower Bound of Negatives | https://proofwiki.org/wiki/Negative_of_Upper_Bound_of_Set_of_Real_Numbers_is_Lower_Bound_of_Negatives | https://proofwiki.org/wiki/Negative_of_Upper_Bound_of_Set_of_Real_Numbers_is_Lower_Bound_of_Negatives | [
"Real Analysis"
] | [
"Definition:Subset",
"Definition:Real Number",
"Definition:Set",
"Definition:Negative of Real Number",
"Definition:Element",
"Definition:Upper Bound of Set/Real Numbers",
"Definition:Lower Bound of Set/Real Numbers"
] | [
"Definition:Set",
"Negative of Lower Bound of Set of Real Numbers is Upper Bound of Negatives",
"Definition:Lower Bound of Set/Real Numbers",
"Definition:Upper Bound of Set/Real Numbers",
"Definition:Lower Bound of Set/Real Numbers",
"Definition:Upper Bound of Set/Real Numbers"
] |
proofwiki-17860 | Combination Theorem for Continuous Functions/Real/Difference Rule | :$f - g$ is continuous on $S$. | We have that:
:$\map {\paren {f - g} } x = \map {\paren {f + \paren {-g} } } x$
From Multiple Rule for Continuous Real Functions:
:$-g$ is continuous on $S$.
From Sum Rule for Continuous Real Functions:
:$f + \paren {-g}$ is continuous on $S$.
The result follows.
{{qed}} | :$f - g$ is [[Definition:Continuous Real Function|continuous]] on $S$. | We have that:
:$\map {\paren {f - g} } x = \map {\paren {f + \paren {-g} } } x$
From [[Multiple Rule for Continuous Real Functions]]:
:$-g$ is [[Definition:Continuous Real Function|continuous]] on $S$.
From [[Sum Rule for Continuous Real Functions]]:
:$f + \paren {-g}$ is [[Definition:Continuous Real Function|contin... | Combination Theorem for Continuous Functions/Real/Difference Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Functions/Real/Difference_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Functions/Real/Difference_Rule | [
"Combination Theorems for Continuous Real Functions"
] | [
"Definition:Continuous Real Function"
] | [
"Combination Theorem for Continuous Functions/Real/Multiple Rule",
"Definition:Continuous Real Function",
"Combination Theorem for Continuous Functions/Real/Sum Rule",
"Definition:Continuous Real Function"
] |
proofwiki-17861 | Dirichlet Function is Discontinuous | Let $D: \R \to \R$ denote the Dirichlet function:
:$\forall x \in \R: \map D x = \begin {cases} c & : x \in \Q \\ d & : x \notin \Q \end {cases}$
where $\Q$ denotes the set of rational numbers.
Then $D$ is discontinuous at every $x \in \R$. | Let $\epsilon = \dfrac {\size {c - d} } 2$.
Let $x \in \Q$.
Let $\delta \in \R_{>0}$ be arbitrary.
Let $y \in \Q$ such that $\size {x - y} < \delta$.
{{WLOG}}, let $y > x$.
From Between two Rational Numbers exists Irrational Number:
:$\exists z \in \R \setminus \Q: x < z < y$
and so:
:$\size {\map D x - \map D z} = \si... | Let $D: \R \to \R$ denote the [[Definition:Dirichlet Function|Dirichlet function]]:
:$\forall x \in \R: \map D x = \begin {cases} c & : x \in \Q \\ d & : x \notin \Q \end {cases}$
where $\Q$ denotes the set of [[Definition:Rational Number|rational numbers]].
Then $D$ is [[Definition:Discontinuous Real Function at Poi... | Let $\epsilon = \dfrac {\size {c - d} } 2$.
Let $x \in \Q$.
Let $\delta \in \R_{>0}$ be arbitrary.
Let $y \in \Q$ such that $\size {x - y} < \delta$.
{{WLOG}}, let $y > x$.
From [[Between two Rational Numbers exists Irrational Number]]:
:$\exists z \in \R \setminus \Q: x < z < y$
and so:
:$\size {\map D x - \ma... | Dirichlet Function is Discontinuous | https://proofwiki.org/wiki/Dirichlet_Function_is_Discontinuous | https://proofwiki.org/wiki/Dirichlet_Function_is_Discontinuous | [
"Dirichlet Functions"
] | [
"Definition:Dirichlet Function",
"Definition:Rational Number",
"Definition:Discontinuous Mapping/Real Function/Point"
] | [
"Between two Rational Numbers exists Irrational Number",
"Definition:Continuous Real Function/Point",
"Definition:Discontinuous Mapping/Real Function/Point",
"Between two Real Numbers exists Rational Number",
"Definition:Continuous Real Function/Point",
"Definition:Discontinuous Mapping/Real Function/Poin... |
proofwiki-17862 | Thomae Function is Continuous at Irrational Numbers | Let $D_M: \R \to \R$ denote the Thomae function:
:$\forall x \in \R: \map {D_M} x = \begin {cases} 0 & : x = 0 \text { or } x \notin \Q \\ \dfrac 1 q & : x = \dfrac p q : p, q \in \Z, p \perp q, q > 0 \end {cases}$
where:
:$\Q$ denotes the set of rational numbers
:$\Z$ denotes the integers
:$p \perp q$ denotes that $p$... | === Rational $x$ ===
Let $x = \dfrac p q \in \Q \setminus \set 0$ such that $\dfrac p q$ is the canonical form of $x$.
Then we have:
:$\map {D_M} x = \dfrac 1 q$
Let $\epsilon = \dfrac 1 {2 q}$.
Let $\delta \in \R_{>0}$.
Then from Between two Real Numbers exists Irrational Number:
:$\exists z \in \R \setminus \Q: x < ... | Let $D_M: \R \to \R$ denote the [[Definition:Thomae Function|Thomae function]]:
:$\forall x \in \R: \map {D_M} x = \begin {cases} 0 & : x = 0 \text { or } x \notin \Q \\ \dfrac 1 q & : x = \dfrac p q : p, q \in \Z, p \perp q, q > 0 \end {cases}$
where:
:$\Q$ denotes the [[Definition:Rational Number|set of rational num... | === Rational $x$ ===
Let $x = \dfrac p q \in \Q \setminus \set 0$ such that $\dfrac p q$ is the [[Definition:Canonical Form of Rational Number|canonical form]] of $x$.
Then we have:
:$\map {D_M} x = \dfrac 1 q$
Let $\epsilon = \dfrac 1 {2 q}$.
Let $\delta \in \R_{>0}$.
Then from [[Between two Real Numbers exists ... | Thomae Function is Continuous at Irrational Numbers | https://proofwiki.org/wiki/Thomae_Function_is_Continuous_at_Irrational_Numbers | https://proofwiki.org/wiki/Thomae_Function_is_Continuous_at_Irrational_Numbers | [
"Thomae Function"
] | [
"Definition:Thomae Function",
"Definition:Rational Number",
"Definition:Integer",
"Definition:Coprime/Integers",
"Definition:Rational Number",
"Definition:Rational Number/Canonical Form",
"Definition:Continuous Real Function/Point",
"Definition:Irrational Number",
"Definition:Discontinuous Mapping/R... | [
"Definition:Rational Number/Canonical Form",
"Between two Real Numbers exists Irrational Number",
"Definition:Rational Number",
"Definition:Discontinuous Mapping/Real Function/Point",
"Definition:Rational Number",
"Definition:Rational Number/Canonical Form"
] |
proofwiki-17863 | Absolute Value of Continuous Real Function is Continuous | Let $f: \R \to \R$ be a real function.
Let $f$ be continuous at a point $a \in \R$.
Then:
:$\size f$ is continuous at $a$
where:
:$\map {\size f} x$ is defined as $\size {\map f x}$. | Let $\epsilon \in \R_{>0}$ be a positive real number.
Because $f$ is continuous at $a$:
:$\exists \delta \in \R_{>0}: \size {x - a} < \delta \implies \size {\map f x - \map f a} < \epsilon$
From Reverse Triangle Inequality:
:$\size {\size {\map f x} - \size {\map f a} } \le \size {\map f x - \map f a}$
and so:
:$\exist... | Let $f: \R \to \R$ be a [[Definition:Real Function|real function]].
Let $f$ be [[Definition:Continuous Real Function at Point|continuous]] at a point $a \in \R$.
Then:
:$\size f$ is [[Definition:Continuous Real Function at Point|continuous]] at $a$
where:
:$\map {\size f} x$ is defined as $\size {\map f x}$. | Let $\epsilon \in \R_{>0}$ be a [[Definition:Positive Real Number|positive real number]].
Because $f$ is [[Definition:Continuous Real Function at Point|continuous]] at $a$:
:$\exists \delta \in \R_{>0}: \size {x - a} < \delta \implies \size {\map f x - \map f a} < \epsilon$
From [[Reverse Triangle Inequality/Real an... | Absolute Value of Continuous Real Function is Continuous | https://proofwiki.org/wiki/Absolute_Value_of_Continuous_Real_Function_is_Continuous | https://proofwiki.org/wiki/Absolute_Value_of_Continuous_Real_Function_is_Continuous | [
"Continuous Real Functions",
"Absolute Value Function"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Point",
"Definition:Continuous Real Function/Point"
] | [
"Definition:Positive/Real Number",
"Definition:Continuous Real Function/Point",
"Reverse Triangle Inequality/Real and Complex Fields",
"Definition:Continuous Real Function/Point"
] |
proofwiki-17864 | Max Operation on Continuous Real Functions is Continuous | Let $f: \R \to \R$ and $g: \R \to \R$ be real functions.
Let $f$ and $g$ be continuous at a point $a \in \R$.
Let $h: \R \to \R$ be the real function defined as:
:$\map h x := \map \max {\map f x, \map g x}$
Then $h$ is continuous at $a$. | From Max is Half of Sum Plus Absolute Difference
:$\max \set {x, y} = \dfrac 1 2 \paren {x + y + \size {x - y} }$
Hence:
:$\max \set {\map f x, \map g x} = \dfrac 1 2 \paren {\map f x + \map g x + \size {\map f x - \map g x} }$
From Difference Rule for Continuous Real Functions:
:$\map f x - \map g x$ is continuous at ... | Let $f: \R \to \R$ and $g: \R \to \R$ be [[Definition:Real Function|real functions]].
Let $f$ and $g$ be [[Definition:Continuous Real Function at Point|continuous]] at a point $a \in \R$.
Let $h: \R \to \R$ be the [[Definition:Real Function|real function]] defined as:
:$\map h x := \map \max {\map f x, \map g x}$
... | From [[Max is Half of Sum Plus Absolute Difference]]
:$\max \set {x, y} = \dfrac 1 2 \paren {x + y + \size {x - y} }$
Hence:
:$\max \set {\map f x, \map g x} = \dfrac 1 2 \paren {\map f x + \map g x + \size {\map f x - \map g x} }$
From [[Difference Rule for Continuous Real Functions]]:
:$\map f x - \map g x$ is [[De... | Max Operation on Continuous Real Functions is Continuous | https://proofwiki.org/wiki/Max_Operation_on_Continuous_Real_Functions_is_Continuous | https://proofwiki.org/wiki/Max_Operation_on_Continuous_Real_Functions_is_Continuous | [
"Continuous Real Functions",
"Max Operation"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Point",
"Definition:Real Function",
"Definition:Continuous Real Function/Point"
] | [
"Max is Half of Sum Plus Absolute Difference",
"Combination Theorem for Continuous Functions/Real/Difference Rule",
"Definition:Continuous Real Function/Point",
"Absolute Value of Continuous Real Function is Continuous",
"Definition:Continuous Real Function/Point",
"Combination Theorem for Continuous Func... |
proofwiki-17865 | Min Operation on Continuous Real Functions is Continuous | Let $f: \R \to \R$ and $g: \R \to \R$ be real functions.
Let $f$ and $g$ be continuous at a point $a \in \R$.
Let $h: \R \to \R$ be the real function defined as:
:$\map h x := \map \min {\map f x, \map g x}$
Then $h$ is continuous at $a$. | From Min is Half of Sum Less Absolute Difference
:$\min \set{x, y} = \dfrac 1 2 \paren {x + y - \size {x - y} }$
Hence:
:$\min \set {\map f x, \map g x} = \dfrac 1 2 \paren {\map f x + \map g x - \size {\map f x - \map g x} }$
From Difference Rule for Continuous Real Functions:
:$\map f x - \map g x$ is continuous at $... | Let $f: \R \to \R$ and $g: \R \to \R$ be [[Definition:Real Function|real functions]].
Let $f$ and $g$ be [[Definition:Continuous Real Function at Point|continuous]] at a point $a \in \R$.
Let $h: \R \to \R$ be the [[Definition:Real Function|real function]] defined as:
:$\map h x := \map \min {\map f x, \map g x}$
... | From [[Min is Half of Sum Less Absolute Difference]]
:$\min \set{x, y} = \dfrac 1 2 \paren {x + y - \size {x - y} }$
Hence:
:$\min \set {\map f x, \map g x} = \dfrac 1 2 \paren {\map f x + \map g x - \size {\map f x - \map g x} }$
From [[Difference Rule for Continuous Real Functions]]:
:$\map f x - \map g x$ is [[Def... | Min Operation on Continuous Real Functions is Continuous | https://proofwiki.org/wiki/Min_Operation_on_Continuous_Real_Functions_is_Continuous | https://proofwiki.org/wiki/Min_Operation_on_Continuous_Real_Functions_is_Continuous | [
"Continuous Real Functions",
"Min Operation"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Point",
"Definition:Real Function",
"Definition:Continuous Real Function/Point"
] | [
"Min is Half of Sum Less Absolute Difference",
"Combination Theorem for Continuous Functions/Real/Difference Rule",
"Definition:Continuous Real Function/Point",
"Absolute Value of Continuous Real Function is Continuous",
"Definition:Continuous Real Function/Point",
"Combination Theorem for Continuous Func... |
proofwiki-17866 | Power of Maximum is not Greater than Sum of Powers | Let $a_1, a_2, \ldots, a_r$ be non-negative real numbers.
Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $a = \max \set {a_1, a_2, \ldots, a_r}$.
Then:
:$a^n \le a_1^n + a_2^n + \cdots + a_r^n$ | By definition of the $\max$ operation:
:$\exists k \in \set {1, 2, \ldots, r}: a_k = a$
Hence:
{{begin-eqn}}
{{eqn | l = a^n
| r = a_k^n
| c =
}}
{{eqn | o = \le
| r = a_1^n + a_2^n + \cdots + a_k^n + \cdots + a_r^n
| c =
}}
{{end-eqn}}
Hence the result.
{{qed}} | Let $a_1, a_2, \ldots, a_r$ be [[Definition:Non-Negative Real Number|non-negative real numbers]].
Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $a = \max \set {a_1, a_2, \ldots, a_r}$.
Then:
:$a^n \le a_1^n + a_2^n + \cdots + a_r^n$ | By definition of the $\max$ operation:
:$\exists k \in \set {1, 2, \ldots, r}: a_k = a$
Hence:
{{begin-eqn}}
{{eqn | l = a^n
| r = a_k^n
| c =
}}
{{eqn | o = \le
| r = a_1^n + a_2^n + \cdots + a_k^n + \cdots + a_r^n
| c =
}}
{{end-eqn}}
Hence the result.
{{qed}} | Power of Maximum is not Greater than Sum of Powers | https://proofwiki.org/wiki/Power_of_Maximum_is_not_Greater_than_Sum_of_Powers | https://proofwiki.org/wiki/Power_of_Maximum_is_not_Greater_than_Sum_of_Powers | [
"Inequalities"
] | [
"Definition:Positive/Real Number",
"Definition:Strictly Positive/Integer"
] | [] |
proofwiki-17867 | Sum of r Powers is not Greater than r times Power of Maximum | Let $a_1, a_2, \ldots, a_r$ be non-negative real numbers.
Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $a = \max \set {a_1, a_2, \ldots, a_r}$.
Then:
:$a_1^n + a_2^n + \cdots + a_r^n \le r a^n$ | By definition of the $\max$ operation:
:$\exists k \in \set {1, 2, \ldots, r}: a_k = a$
Then:
:$\forall i \in \set {1, 2, \ldots, r}: a_i \le a_k$
Hence:
{{begin-eqn}}
{{eqn | q = \forall i \in \set {1, 2, \ldots, r}
| l = a_i
| r = a_k
| c =
}}
{{eqn | ll= \leadsto
| l = \sum_{i \mathop = 1}^r... | Let $a_1, a_2, \ldots, a_r$ be [[Definition:Non-Negative Real Number|non-negative real numbers]].
Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $a = \max \set {a_1, a_2, \ldots, a_r}$.
Then:
:$a_1^n + a_2^n + \cdots + a_r^n \le r a^n$ | By definition of the $\max$ operation:
:$\exists k \in \set {1, 2, \ldots, r}: a_k = a$
Then:
:$\forall i \in \set {1, 2, \ldots, r}: a_i \le a_k$
Hence:
{{begin-eqn}}
{{eqn | q = \forall i \in \set {1, 2, \ldots, r}
| l = a_i
| r = a_k
| c =
}}
{{eqn | ll= \leadsto
| l = \sum_{i \mathop = ... | Sum of r Powers is not Greater than r times Power of Maximum | https://proofwiki.org/wiki/Sum_of_r_Powers_is_not_Greater_than_r_times_Power_of_Maximum | https://proofwiki.org/wiki/Sum_of_r_Powers_is_not_Greater_than_r_times_Power_of_Maximum | [
"Inequalities"
] | [
"Definition:Positive/Real Number",
"Definition:Strictly Positive/Integer"
] | [] |
proofwiki-17868 | Sum of r Powers is between Power of Maximum and r times Power of Maximum | Let $a_1, a_2, \ldots, a_r$ be non-negative real numbers.
Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $a = \max \set {a_1, a_2, \ldots, a_r}$.
Then:
:$a^n \le a_1^n + a_2^n + \cdots + a_r^n \le r a^n$ | This proof is divided into $2$ parts: | Let $a_1, a_2, \ldots, a_r$ be [[Definition:Non-Negative Real Number|non-negative real numbers]].
Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $a = \max \set {a_1, a_2, \ldots, a_r}$.
Then:
:$a^n \le a_1^n + a_2^n + \cdots + a_r^n \le r a^n$ | This proof is divided into $2$ parts: | Sum of r Powers is between Power of Maximum and r times Power of Maximum | https://proofwiki.org/wiki/Sum_of_r_Powers_is_between_Power_of_Maximum_and_r_times_Power_of_Maximum | https://proofwiki.org/wiki/Sum_of_r_Powers_is_between_Power_of_Maximum_and_r_times_Power_of_Maximum | [
"Inequalities"
] | [
"Definition:Positive/Real Number",
"Definition:Strictly Positive/Integer"
] | [] |
proofwiki-17869 | Limit of nth Root of Sum of nth Powers equals Maximum | Let $a_1, a_2, \ldots, a_r$ be non-negative real numbers.
Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $a = \max \set {a_1, a_2, \ldots, a_r}$.
Then:
:$\ds \lim_{n \mathop \to \infty} \paren {a_1^n + a_2^n + \cdots + a_r^n}^{1 / n} = a$ | From Sum of $r$ Powers is between Power of Maximum and $r$ times Power of Maximum:
:$a^n \le a_1^n + a_2^n + \cdots + a_r^n \le r a^n$
and so:
:$a \le \paren {a_1^n + a_2^n + \cdots + a_r^n}^{1 / n} \le r^{1/n} a$
From Limit of Integer to Reciprocal Power:
:$n^{1 / n} \to 1$ as $n \to \infty$
Then for $n > r$:
:$1 < r^... | Let $a_1, a_2, \ldots, a_r$ be [[Definition:Non-Negative Real Number|non-negative real numbers]].
Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $a = \max \set {a_1, a_2, \ldots, a_r}$.
Then:
:$\ds \lim_{n \mathop \to \infty} \paren {a_1^n + a_2^n + \cdots + a_r^n... | From [[Sum of r Powers is between Power of Maximum and r times Power of Maximum|Sum of $r$ Powers is between Power of Maximum and $r$ times Power of Maximum]]:
:$a^n \le a_1^n + a_2^n + \cdots + a_r^n \le r a^n$
and so:
:$a \le \paren {a_1^n + a_2^n + \cdots + a_r^n}^{1 / n} \le r^{1/n} a$
From [[Limit of Integer to... | Limit of nth Root of Sum of nth Powers equals Maximum | https://proofwiki.org/wiki/Limit_of_nth_Root_of_Sum_of_nth_Powers_equals_Maximum | https://proofwiki.org/wiki/Limit_of_nth_Root_of_Sum_of_nth_Powers_equals_Maximum | [
"Examples of Limits of Real Functions"
] | [
"Definition:Positive/Real Number",
"Definition:Strictly Positive/Integer"
] | [
"Sum of r Powers is between Power of Maximum and r times Power of Maximum",
"Limit of Integer to Reciprocal Power",
"Squeeze Theorem"
] |
proofwiki-17870 | Convergence of Odd and Even Subsequences to Same Limit | Let $\sequence {s_n}$ be a real sequence.
Let the subsequences $\sequence {s_{2 n} }$ and $\sequence {s_{2 n + 1} }$ both converge to the same limit $l$.
Then $\sequence {s_n}$ also converges to the same limit $l$. | Suppose that $\sequence {s_n}$ converge to a limit.
Then from Limit of Subsequence equals Limit of Real Sequence, $\sequence {s_{2 n} }$ and $\sequence {s_{2 n + 1} }$ both converge to the same limit.
They do so converge, and that limit is $l$.
So, if $\sequence {s_n}$ converges, it converges to the limit $l$.
{{AimFor... | Let $\sequence {s_n}$ be a [[Definition:Real Sequence|real sequence]].
Let the [[Definition:Subsequence|subsequences]] $\sequence {s_{2 n} }$ and $\sequence {s_{2 n + 1} }$ both [[Definition:Convergent Real Sequence|converge]] to the same [[Definition:Limit of Real Sequence|limit]] $l$.
Then $\sequence {s_n}$ also [... | Suppose that $\sequence {s_n}$ [[Definition:Convergent Real Sequence|converge]] to a [[Definition:Limit of Real Sequence|limit]].
Then from [[Limit of Subsequence equals Limit of Real Sequence]], $\sequence {s_{2 n} }$ and $\sequence {s_{2 n + 1} }$ both [[Definition:Convergent Real Sequence|converge]] to the same [[D... | Convergence of Odd and Even Subsequences to Same Limit | https://proofwiki.org/wiki/Convergence_of_Odd_and_Even_Subsequences_to_Same_Limit | https://proofwiki.org/wiki/Convergence_of_Odd_and_Even_Subsequences_to_Same_Limit | [
"Subsequences"
] | [
"Definition:Real Sequence",
"Definition:Subsequence",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Limit of Sequence/Real Numbers",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Limit of Sequence/Real Numbers"
] | [
"Definition:Convergent Sequence/Real Numbers",
"Definition:Limit of Sequence/Real Numbers",
"Limit of Subsequence equals Limit of Sequence/Real Numbers",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Limit of Sequence/Real Numbers",
"Definition:Convergent Sequence/Real Numbers",
"Definition... |
proofwiki-17871 | Hilbert Cube is Compact/2-Sequence Space with 2-Norm | Let $\ell^2$ be the 2-sequence space.
Let $\norm {\, \cdot \,}_2$ be the 2-norm.
Let $C$ be the Hilbert cube.
Then $C$ is compact in $\struct {\ell^2, \norm {\, \cdot \,}_2}$. | === Construction of subsequence ===
Let $\sequence {\mathbf x_m}_{m \mathop \in \N}$ be a sequence in $C$.
Let $\mathbf x_m = \tuple {x_m^{\paren n}}_{n \mathop \in \N}$.
By definition of Hilbert cube:
:$\ds \forall n,m \in \N : 0 \le x_m^{\paren n} \le \frac 1 n$
Consider the case $n = 1$.
Then:
:$\ds \forall m \in \N... | Let $\ell^2$ be the [[Definition:P-Sequence Space|2-sequence space]].
Let $\norm {\, \cdot \,}_2$ be the [[Definition:P-Norm|2-norm]].
Let $C$ be the [[Definition:Hilbert Cube|Hilbert cube]].
Then $C$ is [[Definition:Compact Subset of Normed Vector Space|compact]] in [[P-Sequence Space with P-Norm forms Normed Vect... | === Construction of subsequence ===
Let $\sequence {\mathbf x_m}_{m \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $C$.
Let $\mathbf x_m = \tuple {x_m^{\paren n}}_{n \mathop \in \N}$.
By definition of [[Definition:Hilbert Cube|Hilbert cube]]:
:$\ds \forall n,m \in \N : 0 \le x_m^{\paren n} \le \frac 1 n$... | Hilbert Cube is Compact/2-Sequence Space with 2-Norm | https://proofwiki.org/wiki/Hilbert_Cube_is_Compact/2-Sequence_Space_with_2-Norm | https://proofwiki.org/wiki/Hilbert_Cube_is_Compact/2-Sequence_Space_with_2-Norm | [
"Hilbert Cube is Compact"
] | [
"Definition:P-Sequence Space",
"Definition:P-Norm",
"Definition:Hilbert Cube",
"Definition:Compact Space/Normed Vector Space",
"P-Sequence Space with P-Norm forms Normed Vector Space"
] | [
"Definition:Sequence",
"Definition:Hilbert Cube",
"Closed Real Interval is Compact Space",
"Definition:Compact",
"Definition:Subsequence",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Limit of Sequence/Normed Vector Space",
"Definition:Subsequence",
"Definition:Convergent Sequen... |
proofwiki-17872 | Standard Discrete Metric is not Topologically Equivalent to p-Product Metrics | For $n \in \N$, let $\R^n$ be an Euclidean space.
Let $p \in \R_{\ge 1}$.
Let $d_p$ be the $p$-product metric on $\R^n$.
Let $d_0$ be the standard discrete metric on $\R^n$.
Then $d_p$ and $d_0$ are not topologically equivalent. | From Open Ball in Standard Discrete Metric Space it is seen that singletons are open sets in $\struct {\R^n, d_0}$.
However, this is not the case in the $\struct {\R^n, d_p}$.
{{qed}} | For $n \in \N$, let $\R^n$ be an [[Definition:Euclidean Space|Euclidean space]].
Let $p \in \R_{\ge 1}$.
Let $d_p$ be the [[Definition:P-Product Metric on Real Vector Space|$p$-product metric]] on $\R^n$.
Let $d_0$ be the [[Definition:Standard Discrete Metric|standard discrete metric]] on $\R^n$.
Then $d_p$ and $d... | From [[Open Ball in Standard Discrete Metric Space]] it is seen that [[Definition:Singleton|singletons]] are [[Definition:Open Set (Metric Space)|open sets]] in $\struct {\R^n, d_0}$.
However, this is not the case in the $\struct {\R^n, d_p}$.
{{qed}} | Standard Discrete Metric is not Topologically Equivalent to p-Product Metrics | https://proofwiki.org/wiki/Standard_Discrete_Metric_is_not_Topologically_Equivalent_to_p-Product_Metrics | https://proofwiki.org/wiki/Standard_Discrete_Metric_is_not_Topologically_Equivalent_to_p-Product_Metrics | [
"P-Product Metrics",
"Standard Discrete Metric"
] | [
"Definition:Euclidean Space",
"Definition:P-Product Metric/Real Vector Space",
"Definition:Standard Discrete Metric",
"Definition:Topologically Equivalent Metrics"
] | [
"Open Ball in Standard Discrete Metric Space",
"Definition:Singleton",
"Definition:Open Set/Metric Space"
] |
proofwiki-17873 | Supremum Metric and L1 Metric on Closed Real Intervals are not Topologically Equivalent | Let $S$ be the set of all real functions which are continuous on the closed interval $\closedint a b$.
Let $d_1$ be the $L^1$ metric on $S$:
:$\ds \forall f, g \in S: \map {d_1} {f, g} := \int_a^b \size {\map f t - \map g t} \rd t$
Let $d_\infty$ be the supremum metric on $S$:
:$\ds \forall f, g \in S: \map {d_\infty} ... | Let $f, g \in S$.
Then by definition of supremum metric:
:$\forall x \in \closedint a b: \size {\map f x - \map g x} \le \map {d_\infty} {f, g}$
Hence by ...
:$\map {d_1} {f, g} = \ds \int_a^b \size {\map f x - \map g x} \rd x \le \paren {b - a} \map {d_\infty} {f, g}$
{{finish|Find that link}}
and so:
:$\map {B_\epsil... | Let $S$ be the [[Definition:Set|set]] of all [[Definition:Real Function|real functions]] which are [[Definition:Continuous Real Function|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Let $d_1$ be the [[Definition:L1 Metric on Closed Real Interval|$L^1$ metric]] on $S$:
:$\ds... | Let $f, g \in S$.
Then by definition of [[Definition:Supremum Metric on Continuous Real Functions|supremum metric]]:
:$\forall x \in \closedint a b: \size {\map f x - \map g x} \le \map {d_\infty} {f, g}$
Hence by ...
:$\map {d_1} {f, g} = \ds \int_a^b \size {\map f x - \map g x} \rd x \le \paren {b - a} \map {d_\in... | Supremum Metric and L1 Metric on Closed Real Intervals are not Topologically Equivalent | https://proofwiki.org/wiki/Supremum_Metric_and_L1_Metric_on_Closed_Real_Intervals_are_not_Topologically_Equivalent | https://proofwiki.org/wiki/Supremum_Metric_and_L1_Metric_on_Closed_Real_Intervals_are_not_Topologically_Equivalent | [
"L1 Metric",
"Supremum Metric"
] | [
"Definition:Set",
"Definition:Real Function",
"Definition:Continuous Real Function",
"Definition:Real Interval/Closed",
"Definition:L1 Metric/Closed Real Interval",
"Definition:Supremum Metric/Continuous Real Functions",
"Definition:Topologically Equivalent Metrics"
] | [
"Definition:Supremum Metric/Continuous Real Functions",
"Definition:Constant Mapping",
"Definition:Open Set/Metric Space",
"Definition:Open Set/Metric Space",
"Definition:Continuous Real Function",
"Definition:Open Set/Metric Space"
] |
proofwiki-17874 | Metrics on Space are Topologically Equivalent iff Identity Mapping is Homemorphism | Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be metric spaces on the same underlying set $A$.
Let $I_A$ denote the identity mapping on $A$.
Then:
:$d_1$ and $d_2$ are topologically equivalent
{{iff}}:
:$I_A: M_1 \to M_2$ is a homeomorphism. | First we establish the following:
From Identity Mapping is Bijection, we have that $I_A$ is a bijection.
From Inverse of Identity Mapping, $\paren {I_A}^{-1} = I_A$.
By Identity Mapping on Metric Space is Continuous:
:$(1): \quad I_A$ is a $\tuple {d_1, d_1}$-continuous mapping.
:$(2): \quad I_A^{-1} = I_A$ is a $\tupl... | Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be [[Definition:Metric Space|metric spaces]] on the same [[Definition:Underlying Set of Metric Space|underlying set]] $A$.
Let $I_A$ denote the [[Definition:Identity Mapping|identity mapping]] on $A$.
Then:
:$d_1$ and $d_2$ are [[Definition:Topologically Equiv... | First we establish the following:
From [[Identity Mapping is Bijection]], we have that $I_A$ is a [[Definition:Bijection|bijection]].
From [[Inverse of Identity Mapping]], $\paren {I_A}^{-1} = I_A$.
By [[Identity Mapping on Metric Space is Continuous]]:
:$(1): \quad I_A$ is a [[Definition:Continuous Mapping (Metric ... | Metrics on Space are Topologically Equivalent iff Identity Mapping is Homemorphism | https://proofwiki.org/wiki/Metrics_on_Space_are_Topologically_Equivalent_iff_Identity_Mapping_is_Homemorphism | https://proofwiki.org/wiki/Metrics_on_Space_are_Topologically_Equivalent_iff_Identity_Mapping_is_Homemorphism | [
"Topologically Equivalent Metrics"
] | [
"Definition:Metric Space",
"Definition:Underlying Set/Metric Space",
"Definition:Identity Mapping",
"Definition:Topologically Equivalent Metrics",
"Definition:Homeomorphism/Metric Spaces"
] | [
"Identity Mapping is Bijection",
"Definition:Bijection",
"Inverse of Identity Mapping",
"Identity Mapping is Continuous/Metric Space",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Continuous ... |
proofwiki-17875 | Metrics on Space are Lipschitz Equivalent iff Identity Mapping is Lipschitz Equivalence | Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be metric spaces on the same underlying set $A$.
Let $I_A$ denote the identity mapping on $A$.
Then:
:$d_1$ and $d_2$ are Lipschitz equivalent
{{iff}}:
:$I_A: M_1 \to M_2$ is a Lipschitz equivalence. | By definition of identity mapping:
:$\forall x \in A: \map {I_A} x = x$ | Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be [[Definition:Metric Space|metric spaces]] on the same [[Definition:Underlying Set of Metric Space|underlying set]] $A$.
Let $I_A$ denote the [[Definition:Identity Mapping|identity mapping]] on $A$.
Then:
:$d_1$ and $d_2$ are [[Definition:Lipschitz Equivalen... | By definition of [[Definition:Identity Mapping|identity mapping]]:
:$\forall x \in A: \map {I_A} x = x$ | Metrics on Space are Lipschitz Equivalent iff Identity Mapping is Lipschitz Equivalence | https://proofwiki.org/wiki/Metrics_on_Space_are_Lipschitz_Equivalent_iff_Identity_Mapping_is_Lipschitz_Equivalence | https://proofwiki.org/wiki/Metrics_on_Space_are_Lipschitz_Equivalent_iff_Identity_Mapping_is_Lipschitz_Equivalence | [
"Lipschitz Equivalence"
] | [
"Definition:Metric Space",
"Definition:Underlying Set/Metric Space",
"Definition:Identity Mapping",
"Definition:Lipschitz Equivalence/Metrics",
"Definition:Lipschitz Equivalence/Metric Spaces"
] | [
"Definition:Identity Mapping"
] |
proofwiki-17876 | Homeomorphism yields Topologically Equivalent Metric | Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be metric spaces on the same underlying set $A$.
Let $f: M_1 \to M_2$ be a homeomorphism.
Let $d_3: A^2 \to \R$ be a mapping defined as:
:$\map {d_3} {x, y} := \map {d_2} {\map f x, \map f y}$
Then $d_3$ yields a metric on $M_1$ which is topologically equivalent... | {{ProofWanted|Easy when you understand what you're doing}} | Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be [[Definition:Metric Space|metric spaces]] on the same [[Definition:Underlying Set of Metric Space|underlying set]] $A$.
Let $f: M_1 \to M_2$ be a [[Definition:Homeomorphism (Metric Spaces)|homeomorphism]].
Let $d_3: A^2 \to \R$ be a [[Definition:Mapping|ma... | {{ProofWanted|Easy when you understand what you're doing}} | Homeomorphism yields Topologically Equivalent Metric | https://proofwiki.org/wiki/Homeomorphism_yields_Topologically_Equivalent_Metric | https://proofwiki.org/wiki/Homeomorphism_yields_Topologically_Equivalent_Metric | [
"Topologically Equivalent Metrics"
] | [
"Definition:Metric Space",
"Definition:Underlying Set/Metric Space",
"Definition:Homeomorphism/Metric Spaces",
"Definition:Mapping",
"Definition:Metric Space/Metric",
"Definition:Topologically Equivalent Metrics"
] | [] |
proofwiki-17877 | Lipschitz Equivalence yields Lipschitz Equivalent Metric | Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be metric spaces on the same underlying set $A$.
Let $f: M_1 \to M_2$ be a Lipschitz equivalence.
Let $d_3: A^2 \to \R$ be a mapping defined as:
:$\map {d_3} {x, y} := \map {d_2} {\map f x, \map f y}$
Then $d_3$ yields a metric on $M_1$ which is Lipschitz equiva... | {{ProofWanted|Easy when you understand what you're doing}} | Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be [[Definition:Metric Space|metric spaces]] on the same [[Definition:Underlying Set of Metric Space|underlying set]] $A$.
Let $f: M_1 \to M_2$ be a [[Definition:Lipschitz Equivalence (Mapping)|Lipschitz equivalence]].
Let $d_3: A^2 \to \R$ be a [[Definition:... | {{ProofWanted|Easy when you understand what you're doing}} | Lipschitz Equivalence yields Lipschitz Equivalent Metric | https://proofwiki.org/wiki/Lipschitz_Equivalence_yields_Lipschitz_Equivalent_Metric | https://proofwiki.org/wiki/Lipschitz_Equivalence_yields_Lipschitz_Equivalent_Metric | [
"Lipschitz Equivalence"
] | [
"Definition:Metric Space",
"Definition:Underlying Set/Metric Space",
"Definition:Lipschitz Equivalence/Metric Spaces",
"Definition:Mapping",
"Definition:Metric Space/Metric",
"Definition:Lipschitz Equivalence/Metrics"
] | [] |
proofwiki-17878 | Open Balls whose Distance between Centers is Twice Radius are Disjoint | Let $M = \struct {A, d}$ be a metric space.
Let $x, y \in A$ such that $\map d {x, y} = 2 r > 0$.
Let $\map {B_r} x$ denote the open $r$-ball of $x$ in $M$.
Then $\map {B_r} x$ and $\map {B_r} y$ are disjoint. | {{AimForCont}} $\map {B_r} x \cap \map {B_r} y \ne \O$.
Then:
{{begin-eqn}}
{{eqn | q = \exists z \in A
| l = z \in \map {B_r} x
| o = \text { and }
| r = z \in \map {B_r} y
| c = {{Defof|Set Intersection}}
}}
{{eqn | ll= \leadsto
| l = \map d {x, z} < r
| o = \text { and }
| ... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $x, y \in A$ such that $\map d {x, y} = 2 r > 0$.
Let $\map {B_r} x$ denote the [[Definition:Open Ball of Metric Space|open $r$-ball]] of $x$ in $M$.
Then $\map {B_r} x$ and $\map {B_r} y$ are [[Definition:Disjoint Sets|disjoint]]. | {{AimForCont}} $\map {B_r} x \cap \map {B_r} y \ne \O$.
Then:
{{begin-eqn}}
{{eqn | q = \exists z \in A
| l = z \in \map {B_r} x
| o = \text { and }
| r = z \in \map {B_r} y
| c = {{Defof|Set Intersection}}
}}
{{eqn | ll= \leadsto
| l = \map d {x, z} < r
| o = \text { and }
|... | Open Balls whose Distance between Centers is Twice Radius are Disjoint/Proof 1 | https://proofwiki.org/wiki/Open_Balls_whose_Distance_between_Centers_is_Twice_Radius_are_Disjoint | https://proofwiki.org/wiki/Open_Balls_whose_Distance_between_Centers_is_Twice_Radius_are_Disjoint/Proof_1 | [
"Open Balls whose Distance between Centers is Twice Radius are Disjoint",
"Open Balls"
] | [
"Definition:Metric Space",
"Definition:Open Ball",
"Definition:Disjoint Sets"
] | [
"Definition:Contradiction",
"Proof by Contradiction"
] |
proofwiki-17879 | Triangle Inequality/Examples/4 Points | Let $M = \struct {A, d}$ be a metric space.
Let $x, y, z, t \in A$.
Then:
:$\map d {x, z} + \map d {y, t} \ge \size {\map d {x, y} - \map d {z, t} }$ | We have that $\map d {x, z}$, $\map d {y, t}$, $\map d {x, y}$, $\map d {z, t}$ are themselves all real numbers.
Hence the Euclidean metric on the real number line can be applied:
{{begin-eqn}}
{{eqn | l = \size {\map d {x, y} - \map d {z, t} }
| o = \le
| r = \size {\map d {x, y} - \map d {y, z} } + \size ... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $x, y, z, t \in A$.
Then:
:$\map d {x, z} + \map d {y, t} \ge \size {\map d {x, y} - \map d {z, t} }$ | We have that $\map d {x, z}$, $\map d {y, t}$, $\map d {x, y}$, $\map d {z, t}$ are themselves all [[Definition:Real Number|real numbers]].
Hence the [[Definition:Euclidean Metric on Real Number Line|Euclidean metric]] on the [[Definition:Real Number Line|real number line]] can be applied:
{{begin-eqn}}
{{eqn | l = \... | Triangle Inequality/Examples/4 Points | https://proofwiki.org/wiki/Triangle_Inequality/Examples/4_Points | https://proofwiki.org/wiki/Triangle_Inequality/Examples/4_Points | [
"Examples of Triangle Inequality"
] | [
"Definition:Metric Space"
] | [
"Definition:Real Number",
"Definition:Euclidean Metric/Real Number Line",
"Definition:Real Number/Real Number Line",
"Reverse Triangle Inequality"
] |
proofwiki-17880 | Subset of Bounded Subset of Metric Space is Bounded | Let $M = \struct {A, d}$ be a metric space.
Let $B$ be a bounded subset of $M$.
Let $\map \diam B$ denote the diameter of $B$.
Let $C \subseteq B$ be a subset of $B$.
Then $C$ is a bounded subset of $M$ such that:
:$\map \diam C \le \map \diam B$ | {{begin-eqn}}
{{eqn | q = \forall x, y \in B
| l = \map d {x, y}
| o = \le
| r = \map \diam B
| c = {{Defof|Diameter of Subset of Metric Space}}
}}
{{eqn | ll= \leadsto
| q = \forall x, y \in C
| l = \map d {x, y}
| o = \le
| r = \map \diam B
| c = {{Defof|Subset}}
... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $B$ be a [[Definition:Bounded Metric Space|bounded]] [[Definition:Subset|subset]] of $M$.
Let $\map \diam B$ denote the [[Definition:Diameter of Subset of Metric Space|diameter]] of $B$.
Let $C \subseteq B$ be a [[Definition:Subset|subset]]... | {{begin-eqn}}
{{eqn | q = \forall x, y \in B
| l = \map d {x, y}
| o = \le
| r = \map \diam B
| c = {{Defof|Diameter of Subset of Metric Space}}
}}
{{eqn | ll= \leadsto
| q = \forall x, y \in C
| l = \map d {x, y}
| o = \le
| r = \map \diam B
| c = {{Defof|Subset}}
... | Subset of Bounded Subset of Metric Space is Bounded | https://proofwiki.org/wiki/Subset_of_Bounded_Subset_of_Metric_Space_is_Bounded | https://proofwiki.org/wiki/Subset_of_Bounded_Subset_of_Metric_Space_is_Bounded | [
"Bounded Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Bounded Metric Space",
"Definition:Subset",
"Definition:Diameter of Subset of Metric Space",
"Definition:Subset",
"Definition:Bounded Metric Space",
"Definition:Subset"
] | [] |
proofwiki-17881 | Union of Non-Disjoint Bounded Subsets of Metric Space is Bounded | Let $M = \struct {A, d}$ be a metric space.
Let $B$ and $C$ be bounded subsets of $M$ such that $B \cap C \ne \O$.
Let $\map \diam B$ and $\map \diam C$ denote the diameters of $B$ and $C$.
Then $B \cup C$ is a bounded subset of $M$ such that:
:$\map \diam {B \cup C} \le \map \diam B + \map \diam C$ | {{AimForCont}} there exists $x, y \in B \cup C$ such that:
:$\map d {x, y} > \map \diam B + \map \diam C$
$x$ and $y$ cannot both be in $B$ or $C$, otherwise either $\map d {x, y} \le \map \diam B$ or $\map d {x, y} \le \map \diam C$.
{{WLOG}}, let $x \in B$ and $y \in C$.
Let $z \in B \cap C$.
This is possible because... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $B$ and $C$ be [[Definition:Bounded Metric Space|bounded]] [[Definition:Subset|subsets]] of $M$ such that $B \cap C \ne \O$.
Let $\map \diam B$ and $\map \diam C$ denote the [[Definition:Diameter of Subset of Metric Space|diameters]] of $B$ a... | {{AimForCont}} there exists $x, y \in B \cup C$ such that:
:$\map d {x, y} > \map \diam B + \map \diam C$
$x$ and $y$ cannot both be in $B$ or $C$, otherwise either $\map d {x, y} \le \map \diam B$ or $\map d {x, y} \le \map \diam C$.
{{WLOG}}, let $x \in B$ and $y \in C$.
Let $z \in B \cap C$.
This is possible bec... | Union of Non-Disjoint Bounded Subsets of Metric Space is Bounded | https://proofwiki.org/wiki/Union_of_Non-Disjoint_Bounded_Subsets_of_Metric_Space_is_Bounded | https://proofwiki.org/wiki/Union_of_Non-Disjoint_Bounded_Subsets_of_Metric_Space_is_Bounded | [
"Bounded Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Bounded Metric Space",
"Definition:Subset",
"Definition:Diameter of Subset of Metric Space",
"Definition:Bounded Metric Space",
"Definition:Subset"
] | [
"Definition:Contradiction"
] |
proofwiki-17882 | Multiple of Metric forms Metric | Let $M = \struct {A, d}$ be a metric space.
Let $d_1: A^2 \to \R$ be the mapping defined as:
:$\forall \tuple {x, y} \in A^2: \map {d_1} {x, y} = k \map d {x, y}$
for some strictly positive $k \in \R_{>0}$.
Then $d_1$ is a metric for $A$. | It is to be demonstrated that $d_1$ satisfies all the metric space axioms. | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $d_1: A^2 \to \R$ be the [[Definition:Mapping|mapping]] defined as:
:$\forall \tuple {x, y} \in A^2: \map {d_1} {x, y} = k \map d {x, y}$
for some [[Definition:Strictly Positive Real Number|strictly positive]] $k \in \R_{>0}$.
Then $d_1$ is ... | It is to be demonstrated that $d_1$ satisfies all the [[Axiom:Metric Space Axioms|metric space axioms]]. | Multiple of Metric forms Metric | https://proofwiki.org/wiki/Multiple_of_Metric_forms_Metric | https://proofwiki.org/wiki/Multiple_of_Metric_forms_Metric | [
"Examples of Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Mapping",
"Definition:Strictly Positive/Real Number",
"Definition:Metric Space/Metric"
] | [
"Axiom:Metric Space Axioms",
"Axiom:Metric Space Axioms"
] |
proofwiki-17883 | Standard Bounded Metric is Metric | Let $M = \struct {A, d}$ be a metric space.
Let $\bar d: A^2 \to \R$ be the standard bounded metric of $d$:
:$\forall \tuple {x, y} \in A^2: \map {\bar d} {x, y} = \min \set {1, \map d {x, y} }$
Then $\bar d$ is a metric for $A$. | It is to be demonstrated that $\bar d$ satisfies all the metric space axioms. | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $\bar d: A^2 \to \R$ be the [[Definition:Standard Bounded Metric|standard bounded metric]] of $d$:
:$\forall \tuple {x, y} \in A^2: \map {\bar d} {x, y} = \min \set {1, \map d {x, y} }$
Then $\bar d$ is a [[Definition:Metric|metric]] for $A$... | It is to be demonstrated that $\bar d$ satisfies all the [[Axiom:Metric Space Axioms|metric space axioms]]. | Standard Bounded Metric is Metric | https://proofwiki.org/wiki/Standard_Bounded_Metric_is_Metric | https://proofwiki.org/wiki/Standard_Bounded_Metric_is_Metric | [
"Examples of Metric Spaces",
"Standard Bounded Metric is Metric"
] | [
"Definition:Metric Space",
"Definition:Standard Bounded Metric",
"Definition:Metric Space/Metric"
] | [
"Axiom:Metric Space Axioms",
"Axiom:Metric Space Axioms"
] |
proofwiki-17884 | Metric over 1 plus Metric forms Metric | Let $M = \struct {A, d}$ be a metric space.
Let $d_3: A^2 \to \R$ be the mapping defined as:
:$\forall \tuple {x, y} \in A^2: \map {d_3} {x, y} = \dfrac {\map d {x, y} } {1 + \map d {x, y} }$
Then $d_3$ is a metric for $A$. | It is to be demonstrated that $d_3$ satisfies all the metric space axioms. | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $d_3: A^2 \to \R$ be the [[Definition:Mapping|mapping]] defined as:
:$\forall \tuple {x, y} \in A^2: \map {d_3} {x, y} = \dfrac {\map d {x, y} } {1 + \map d {x, y} }$
Then $d_3$ is a [[Definition:Metric|metric]] for $A$. | It is to be demonstrated that $d_3$ satisfies all the [[Axiom:Metric Space Axioms|metric space axioms]]. | Metric over 1 plus Metric forms Metric | https://proofwiki.org/wiki/Metric_over_1_plus_Metric_forms_Metric | https://proofwiki.org/wiki/Metric_over_1_plus_Metric_forms_Metric | [
"Examples of Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Mapping",
"Definition:Metric Space/Metric"
] | [
"Axiom:Metric Space Axioms",
"Axiom:Metric Space Axioms"
] |
proofwiki-17885 | Square of Metric does not necessarily form Metric | Let $M = \struct {A, d}$ be a metric space.
Let $d_4: A^2 \to \R$ be the mapping defined as:
:$\forall \tuple {x, y} \in A^2: \map {d_4} {x, y} = \paren {\map d {x, y} }^2$
Then $d_4$ may or may not be a metric for $A$. | Let $d$ be the standard discrete metric on $M$.
Then:
:$\forall \tuple {x, y} \in A^2: \map {d_4} {x, y} = \map d {x, y}$
and indeed in this case $d_4$ is a metric for $A$.
{{qed|lemma}}
Let $M = \struct {\R, d}$ be the real number line with the usual (Euclidean) metric.
Let $x = 1$, $y = \dfrac 1 2$ and $z = 0$.
We ha... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $d_4: A^2 \to \R$ be the [[Definition:Mapping|mapping]] defined as:
:$\forall \tuple {x, y} \in A^2: \map {d_4} {x, y} = \paren {\map d {x, y} }^2$
Then $d_4$ may or may not be a [[Definition:Metric|metric]] for $A$. | Let $d$ be the [[Definition:Standard Discrete Metric|standard discrete metric]] on $M$.
Then:
:$\forall \tuple {x, y} \in A^2: \map {d_4} {x, y} = \map d {x, y}$
and indeed in this case $d_4$ is a [[Definition:Metric|metric]] for $A$.
{{qed|lemma}}
Let $M = \struct {\R, d}$ be the [[Definition:Real Number Line with... | Square of Metric does not necessarily form Metric | https://proofwiki.org/wiki/Square_of_Metric_does_not_necessarily_form_Metric | https://proofwiki.org/wiki/Square_of_Metric_does_not_necessarily_form_Metric | [
"Examples of Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Mapping",
"Definition:Metric Space/Metric"
] | [
"Definition:Standard Discrete Metric",
"Definition:Metric Space/Metric",
"Definition:Euclidean Metric/Real Number Line"
] |
proofwiki-17886 | Arc Length on Circle forms Metric | Let $A \subseteq \R^2$ be the set defined as:
:$A = \set {\tuple {x_1, x_2}: x_1^2 + y_2^2 = 1}$
Thus from Equation of Unit Circle, $A$ is the unit circle embedded in the Cartesian plane.
Let $d: A^2 \to \R$ be the mapping defined as:
:$\forall \tuple {x, y} \in A^2: \map d {x, y} = \begin {cases} 0 & : x = y \\ \pi & ... | It is to be demonstrated that $d$ satisfies all the metric space axioms. | Let $A \subseteq \R^2$ be the [[Definition:Set|set]] defined as:
:$A = \set {\tuple {x_1, x_2}: x_1^2 + y_2^2 = 1}$
Thus from [[Equation of Unit Circle]], $A$ is the [[Definition:Unit Circle|unit circle]] embedded in the [[Definition:Cartesian Plane|Cartesian plane]].
Let $d: A^2 \to \R$ be the [[Definition:Mapping|ma... | It is to be demonstrated that $d$ satisfies all the [[Axiom:Metric Space Axioms|metric space axioms]]. | Arc Length on Circle forms Metric | https://proofwiki.org/wiki/Arc_Length_on_Circle_forms_Metric | https://proofwiki.org/wiki/Arc_Length_on_Circle_forms_Metric | [
"Examples of Metric Spaces"
] | [
"Definition:Set",
"Equation of Unit Circle",
"Definition:Unit Circle",
"Definition:Cartesian Plane",
"Definition:Mapping",
"Definition:Arc Length",
"Definition:Circle/Arc/Minor",
"Definition:Metric Space/Metric"
] | [
"Axiom:Metric Space Axioms",
"Axiom:Metric Space Axioms"
] |
proofwiki-17887 | Catesian Product of Open Real Intervals is Open in Real Number Plane | Let $\struct {\R^2, d}$ denote the real number plane with the Euclidean metric.
Let $\openint a b$ and $\openint c d$ be open real intervals.
Then their Cartesian product:
:$S = \openint a b \times \openint c d$
is an open set of $\struct {\R^2, d}$. | Let $P = \tuple {x, y} \in S$.
Then by definition:
:$a < x < b$
and:
:$c < y < d$
Let $\epsilon = \min \set {x - a, b - x, y - c, d - y}$
By definition, $\epsilon > 0$.
Consider the open ball $\map {B_\epsilon} P$.
Let $Q = \tuple {x_0, y_0} \in \map {B_\epsilon} P$ be an arbitrary point in $\map {B_\epsilon} P$.
By de... | Let $\struct {\R^2, d}$ denote the [[Definition:Real Number Plane with Euclidean Metric|real number plane with the Euclidean metric]].
Let $\openint a b$ and $\openint c d$ be [[Definition:Open Real Interval|open real intervals]].
Then their [[Definition:Cartesian Product|Cartesian product]]:
:$S = \openint a b \tim... | Let $P = \tuple {x, y} \in S$.
Then by definition:
:$a < x < b$
and:
:$c < y < d$
Let $\epsilon = \min \set {x - a, b - x, y - c, d - y}$
By definition, $\epsilon > 0$.
Consider the [[Definition:Open Ball of Metric Space|open ball]] $\map {B_\epsilon} P$.
Let $Q = \tuple {x_0, y_0} \in \map {B_\epsilon} P$ be an a... | Catesian Product of Open Real Intervals is Open in Real Number Plane | https://proofwiki.org/wiki/Catesian_Product_of_Open_Real_Intervals_is_Open_in_Real_Number_Plane | https://proofwiki.org/wiki/Catesian_Product_of_Open_Real_Intervals_is_Open_in_Real_Number_Plane | [
"Open Sets"
] | [
"Definition:Euclidean Metric/Real Number Plane",
"Definition:Real Interval/Open",
"Definition:Cartesian Product",
"Definition:Open Set/Metric Space"
] | [
"Definition:Open Ball",
"Definition:Point",
"Definition:Open Set/Metric Space"
] |
proofwiki-17888 | Non-Zero Value of Continuous Real-Valued Function has Neighborhood not including Zero | Let $M = \struct {A, d}$ be a metric space.
Let $f: M \to \R$ be a continuous real-valued function.
Let $\map f a > 0$ for some $a \in M$.
Then there exists $\delta \in \R_{>0}$ such that:
:$\forall x \in \map {B_\delta} a$
where $\map {B_\delta} a$ denotes the open $\delta$-ball of $a$ in $M$. | Let $\epsilon \in \R_{>0}$ such that $\epsilon < \map f a$.
Let $\map {N_\epsilon} {\map f a} = \openint {\map f a - \epsilon} {\map f a + \epsilon}$ be the $\epsilon$-neighborhood of $\map f a$.
Then $\map {N_\epsilon} {\map f a}$ is an open interval of $\map f a$ such that:
:$\forall y \in \map {N_\epsilon} {\map f a... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $f: M \to \R$ be a [[Definition:Continuous Real-Valued Function|continuous real-valued function]].
Let $\map f a > 0$ for some $a \in M$.
Then there exists $\delta \in \R_{>0}$ such that:
:$\forall x \in \map {B_\delta} a$
where $\map {B_\d... | Let $\epsilon \in \R_{>0}$ such that $\epsilon < \map f a$.
Let $\map {N_\epsilon} {\map f a} = \openint {\map f a - \epsilon} {\map f a + \epsilon}$ be the [[Definition:Epsilon-Neighborhood (Real Number Line)|$\epsilon$-neighborhood of $\map f a$]].
Then $\map {N_\epsilon} {\map f a}$ is an [[Definition:Open Interva... | Non-Zero Value of Continuous Real-Valued Function has Neighborhood not including Zero | https://proofwiki.org/wiki/Non-Zero_Value_of_Continuous_Real-Valued_Function_has_Neighborhood_not_including_Zero | https://proofwiki.org/wiki/Non-Zero_Value_of_Continuous_Real-Valued_Function_has_Neighborhood_not_including_Zero | [
"Continuous Mappings on Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Continuous Real-Valued Vector Function",
"Definition:Open Ball"
] | [
"Definition:Neighborhood (Real Analysis)/Epsilon",
"Definition:Interval/Ordered Set/Open",
"Definition:Continuous Real-Valued Vector Function",
"Definition:Open Set/Metric Space",
"Definition:Open Set/Metric Space"
] |
proofwiki-17889 | Combination Theorem for Continuous Mappings/Metric Space/Sum Rule | :$f + g$ is continuous on $M$. | By definition of continuous:
:$\forall a \in M: \ds \lim_{x \mathop \to a} \map f x = \map f a$
:$\forall a \in M: \ds \lim_{x \mathop \to a} \map g x = \map g a$
Let $f$ and $g$ tend to the following limits:
:$\ds \lim_{x \mathop \to a} \map f x = l$
:$\ds \lim_{x \mathop \to a} \map g x = m$
From the Sum Rule for Lim... | :$f + g$ is [[Definition:Continuous Mapping (Metric Space)|continuous]] on $M$. | By definition of [[Definition:Continuous Mapping (Metric Space)|continuous]]:
:$\forall a \in M: \ds \lim_{x \mathop \to a} \map f x = \map f a$
:$\forall a \in M: \ds \lim_{x \mathop \to a} \map g x = \map g a$
Let $f$ and $g$ tend to the following [[Definition:Limit of Real Function|limits]]:
:$\ds \lim_{x \mathop ... | Combination Theorem for Continuous Mappings/Metric Space/Sum Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Metric_Space/Sum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Metric_Space/Sum_Rule | [
"Combination Theorem for Continuous Mappings"
] | [
"Definition:Continuous Mapping (Metric Space)"
] | [
"Definition:Continuous Mapping (Metric Space)",
"Definition:Limit of Real Function",
"Combination Theorem for Limits of Functions/Real/Sum Rule",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Continuous Mapping (Metric Space)"
] |
proofwiki-17890 | Combination Theorem for Limits of Mappings/Metric Space/Sum Rule | :$\ds \lim_{x \mathop \to a} \paren {\map f x + \map g x} = l + m$ | Let $\sequence {x_n}$ be any sequence of elements of $S$ such that:
:$\forall n \in \N_{>0}: x_n \ne a$
:$\ds \lim_{n \mathop \to \infty} \ x_n = a$
By Limit of Function by Convergent Sequences:
:$\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$
:$\ds \lim_{n \mathop \to \infty} \map g {x_n} = m$
By the Sum Rule for R... | :$\ds \lim_{x \mathop \to a} \paren {\map f x + \map g x} = l + m$ | Let $\sequence {x_n}$ be any [[Definition:Sequence|sequence]] of [[Definition:Element|elements]] of $S$ such that:
:$\forall n \in \N_{>0}: x_n \ne a$
:$\ds \lim_{n \mathop \to \infty} \ x_n = a$
By [[Limit of Function by Convergent Sequences]]:
:$\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$
:$\ds \lim_{n \matho... | Combination Theorem for Limits of Mappings/Metric Space/Sum Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Limits_of_Mappings/Metric_Space/Sum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Limits_of_Mappings/Metric_Space/Sum_Rule | [
"Combination Theorem for Limits of Mappings in Metric Spaces"
] | [] | [
"Definition:Sequence",
"Definition:Element",
"Limit of Function by Convergent Sequences",
"Combination Theorem for Sequences/Real/Sum Rule",
"Limit of Function by Convergent Sequences",
"Category:Combination Theorem for Limits of Mappings in Metric Spaces"
] |
proofwiki-17891 | Combination Theorem for Limits of Mappings/Metric Space/Multiple Rule | :$\ds \lim_{x \mathop \to a} \lambda \map f x = \lambda l$ | Let $\sequence {x_n}$ be any sequence of elements of $S$ such that:
:$\forall n \in \N_{>0}: x_n \ne a$
:$\ds \lim_{n \mathop \to \infty} \ x_n = a$
By Limit of Function by Convergent Sequences:
:$\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$
By the Multiple Rule for Real Sequences:
:$\ds \lim_{n \mathop \to \infty... | :$\ds \lim_{x \mathop \to a} \lambda \map f x = \lambda l$ | Let $\sequence {x_n}$ be any [[Definition:Sequence|sequence]] of [[Definition:Element|elements]] of $S$ such that:
:$\forall n \in \N_{>0}: x_n \ne a$
:$\ds \lim_{n \mathop \to \infty} \ x_n = a$
By [[Limit of Function by Convergent Sequences]]:
:$\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$
By the [[Multiple ... | Combination Theorem for Limits of Mappings/Metric Space/Multiple Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Limits_of_Mappings/Metric_Space/Multiple_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Limits_of_Mappings/Metric_Space/Multiple_Rule | [
"Combination Theorem for Limits of Mappings in Metric Spaces"
] | [] | [
"Definition:Sequence",
"Definition:Element",
"Limit of Function by Convergent Sequences",
"Combination Theorem for Sequences/Real/Multiple Rule",
"Limit of Function by Convergent Sequences",
"Category:Combination Theorem for Limits of Mappings in Metric Spaces"
] |
proofwiki-17892 | Combination Theorem for Limits of Mappings/Metric Space/Product Rule | :$\ds \lim_{x \mathop \to a} \paren {\map f x \map g x} = l m$ | Let $\sequence {x_n}$ be any sequence of elements of $S$ such that:
:$\forall n \in \N_{>0}: x_n \ne a$
:$\ds \lim_{n \mathop \to \infty} \ x_n = a$
By Limit of Function by Convergent Sequences:
:$\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$
:$\ds \lim_{n \mathop \to \infty} \map g {x_n} = m$
By the Product Rule f... | :$\ds \lim_{x \mathop \to a} \paren {\map f x \map g x} = l m$ | Let $\sequence {x_n}$ be any [[Definition:Sequence|sequence]] of [[Definition:Element|elements]] of $S$ such that:
:$\forall n \in \N_{>0}: x_n \ne a$
:$\ds \lim_{n \mathop \to \infty} \ x_n = a$
By [[Limit of Function by Convergent Sequences]]:
:$\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$
:$\ds \lim_{n \matho... | Combination Theorem for Limits of Mappings/Metric Space/Product Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Limits_of_Mappings/Metric_Space/Product_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Limits_of_Mappings/Metric_Space/Product_Rule | [
"Combination Theorem for Limits of Mappings in Metric Spaces"
] | [] | [
"Definition:Sequence",
"Definition:Element",
"Limit of Function by Convergent Sequences",
"Combination Theorem for Sequences/Real/Product Rule",
"Limit of Function by Convergent Sequences",
"Category:Combination Theorem for Limits of Mappings in Metric Spaces"
] |
proofwiki-17893 | Combination Theorem for Limits of Mappings/Metric Space/Quotient Rule | :$\ds \lim_{x \mathop \to a} \frac {\map f x} {\map g x} = \frac l m$
provided that $m \ne 0$. | Let $\sequence {x_n}$ be any sequence of elements of $S$ such that:
:$\forall n \in \N_{>0}: x_n \ne a$
:$\ds \lim_{n \mathop \to \infty} \ x_n = a$
By Limit of Function by Convergent Sequences:
:$\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$
:$\ds \lim_{n \mathop \to \infty} \map g {x_n} = m$
By the Quotient Rule ... | :$\ds \lim_{x \mathop \to a} \frac {\map f x} {\map g x} = \frac l m$
provided that $m \ne 0$. | Let $\sequence {x_n}$ be any [[Definition:Sequence|sequence]] of [[Definition:Element|elements]] of $S$ such that:
:$\forall n \in \N_{>0}: x_n \ne a$
:$\ds \lim_{n \mathop \to \infty} \ x_n = a$
By [[Limit of Function by Convergent Sequences]]:
:$\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$
:$\ds \lim_{n \matho... | Combination Theorem for Limits of Mappings/Metric Space/Quotient Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Limits_of_Mappings/Metric_Space/Quotient_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Limits_of_Mappings/Metric_Space/Quotient_Rule | [
"Combination Theorem for Limits of Mappings in Metric Spaces"
] | [] | [
"Definition:Sequence",
"Definition:Element",
"Limit of Function by Convergent Sequences",
"Combination Theorem for Sequences/Real/Quotient Rule",
"Limit of Function by Convergent Sequences",
"Category:Combination Theorem for Limits of Mappings in Metric Spaces"
] |
proofwiki-17894 | Combination Theorem for Continuous Mappings/Metric Space/Difference Rule | :$f - g$ is continuous on $M$. | :$\map {\paren {f - g} } x = \map {\paren {f + \paren {-g} } } x$
From Multiple Rule for Continuous Mappings on Metric Space:
:$-g$ is continuous on $M$.
From Sum Rule for Continuous Mappings on Metric Space:
:$f + \paren {-g}$ is continuous on $M$.
The result follows.
{{qed}} | :$f - g$ is [[Definition:Continuous Mapping (Metric Space)|continuous]] on $M$. | :$\map {\paren {f - g} } x = \map {\paren {f + \paren {-g} } } x$
From [[Multiple Rule for Continuous Mappings on Metric Space]]:
:$-g$ is [[Definition:Continuous Mapping (Metric Space)|continuous]] on $M$.
From [[Sum Rule for Continuous Mappings on Metric Space]]:
:$f + \paren {-g}$ is [[Definition:Continuous Mappin... | Combination Theorem for Continuous Mappings/Metric Space/Difference Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Metric_Space/Difference_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Metric_Space/Difference_Rule | [
"Combination Theorem for Continuous Mappings"
] | [
"Definition:Continuous Mapping (Metric Space)"
] | [
"Combination Theorem for Continuous Mappings/Metric Space/Multiple Rule",
"Definition:Continuous Mapping (Metric Space)",
"Combination Theorem for Continuous Mappings/Metric Space/Sum Rule",
"Definition:Continuous Mapping (Metric Space)"
] |
proofwiki-17895 | Combination Theorem for Continuous Mappings/Metric Space/Multiple Rule | :$\lambda f$ is continuous on $M$. | By definition of continuous:
:$\forall a \in M: \ds \lim_{x \mathop \to a} \map f x = \map f a$
Let $f$ tend to the following limit:
:$\ds \lim_{x \mathop \to a} \map f x = l$
From the Multiple Rule for Limits of Real Functions, we have that:
:$\ds \lim_{x \mathop \to a} \paren {\lambda \map f x} = \lambda l$
So, by de... | :$\lambda f$ is [[Definition:Continuous Mapping (Metric Space)|continuous]] on $M$. | By definition of [[Definition:Continuous Mapping (Metric Space)|continuous]]:
:$\forall a \in M: \ds \lim_{x \mathop \to a} \map f x = \map f a$
Let $f$ tend to the following [[Definition:Limit of Real Function|limit]]:
:$\ds \lim_{x \mathop \to a} \map f x = l$
From the [[Multiple Rule for Limits of Real Functions... | Combination Theorem for Continuous Mappings/Metric Space/Multiple Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Metric_Space/Multiple_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Metric_Space/Multiple_Rule | [
"Combination Theorem for Continuous Mappings"
] | [
"Definition:Continuous Mapping (Metric Space)"
] | [
"Definition:Continuous Mapping (Metric Space)",
"Definition:Limit of Real Function",
"Combination Theorem for Limits of Functions/Real/Multiple Rule",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Continuous Mapping (Metric Space)"
] |
proofwiki-17896 | Combination Theorem for Continuous Mappings/Metric Space/Absolute Value Rule | :$\size f$ is continuous at $a$
where:
:$\map {\size f} x$ is defined as $\size {\map f x}$. | Let $\epsilon \in \R_{>0}$ be a positive real number.
Because $f$ is continuous at $a$:
:$\exists \delta \in \R_{>0}: \map d {x, a} < \delta \implies \size {\map f x - \map f a} < \epsilon$
From Reverse Triangle Inequality:
:$\size {\size {\map f x} - \size {\map f a} } \le \size {\map f x - \map f a}$
and so:
:$\exist... | :$\size f$ is [[Definition:Continuous Real Function at Point|continuous]] at $a$
where:
:$\map {\size f} x$ is defined as $\size {\map f x}$. | Let $\epsilon \in \R_{>0}$ be a [[Definition:Positive Real Number|positive real number]].
Because $f$ is [[Definition:Continuous at Point of Metric Space|continuous]] at $a$:
:$\exists \delta \in \R_{>0}: \map d {x, a} < \delta \implies \size {\map f x - \map f a} < \epsilon$
From [[Reverse Triangle Inequality/Real ... | Combination Theorem for Continuous Mappings/Metric Space/Absolute Value Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Metric_Space/Absolute_Value_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Metric_Space/Absolute_Value_Rule | [
"Combination Theorem for Continuous Mappings"
] | [
"Definition:Continuous Real Function/Point"
] | [
"Definition:Positive/Real Number",
"Definition:Continuous Mapping (Metric Space)/Point",
"Reverse Triangle Inequality/Real and Complex Fields",
"Definition:Continuous Mapping (Metric Space)/Point"
] |
proofwiki-17897 | Combination Theorem for Continuous Mappings/Metric Space/Maximum Rule | :$\max \set {f, g}$ is continuous on $M$. | Let $a \in M$ be arbitrary.
From Max is Half of Sum Plus Absolute Difference
:$\max \set {x, y} = \dfrac 1 2 \paren {x + y + \size {x - y} }$
Hence:
:$\max \set {\map f x, \map g x} = \dfrac 1 2 \paren {\map f x + \map g x + \size {\map f x - \map g x} }$
From Difference Rule for Continuous Mappings on Metric Space:
:$... | :$\max \set {f, g}$ is [[Definition:Continuous Mapping (Metric Space)|continuous]] on $M$. | Let $a \in M$ be arbitrary.
From [[Max is Half of Sum Plus Absolute Difference]]
:$\max \set {x, y} = \dfrac 1 2 \paren {x + y + \size {x - y} }$
Hence:
:$\max \set {\map f x, \map g x} = \dfrac 1 2 \paren {\map f x + \map g x + \size {\map f x - \map g x} }$
From [[Difference Rule for Continuous Mappings on Metric ... | Combination Theorem for Continuous Mappings/Metric Space/Maximum Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Metric_Space/Maximum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Metric_Space/Maximum_Rule | [
"Combination Theorem for Continuous Mappings"
] | [
"Definition:Continuous Mapping (Metric Space)"
] | [
"Max is Half of Sum Plus Absolute Difference",
"Combination Theorem for Continuous Mappings/Metric Space/Difference Rule",
"Definition:Continuous Mapping (Metric Space)/Point",
"Combination Theorem for Continuous Mappings/Metric Space/Absolute Value Rule",
"Definition:Continuous Mapping (Metric Space)/Point... |
proofwiki-17898 | Combination Theorem for Continuous Mappings/Metric Space/Minimum Rule | :$\min \set {f, g}$ is continuous on $M$. | Let $a \in M$ be arbitrary.
From Min is Half of Sum Less Absolute Difference
:$\min \set {x, y} = \dfrac 1 2 \paren {x + y - \size {x - y} }$
Hence:
:$\min \set {\map f x, \map g x} = \dfrac 1 2 \paren {\map f x + \map g x - \size {\map f x - \map g x} }$
From Difference Rule for Continuous Mappings on Metric Space:
:$... | :$\min \set {f, g}$ is [[Definition:Continuous Mapping (Metric Space)|continuous]] on $M$. | Let $a \in M$ be arbitrary.
From [[Min is Half of Sum Less Absolute Difference]]
:$\min \set {x, y} = \dfrac 1 2 \paren {x + y - \size {x - y} }$
Hence:
:$\min \set {\map f x, \map g x} = \dfrac 1 2 \paren {\map f x + \map g x - \size {\map f x - \map g x} }$
From [[Difference Rule for Continuous Mappings on Metric ... | Combination Theorem for Continuous Mappings/Metric Space/Minimum Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Metric_Space/Minimum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Metric_Space/Minimum_Rule | [
"Combination Theorem for Continuous Mappings"
] | [
"Definition:Continuous Mapping (Metric Space)"
] | [
"Min is Half of Sum Less Absolute Difference",
"Combination Theorem for Continuous Mappings/Metric Space/Difference Rule",
"Definition:Continuous Mapping (Metric Space)/Point",
"Combination Theorem for Continuous Mappings/Metric Space/Absolute Value Rule",
"Definition:Continuous Mapping (Metric Space)/Point... |
proofwiki-17899 | Combination Theorem for Continuous Mappings/Metric Space/Combined Sum Rule | :$\lambda f + \mu g$ is continuous on $M$. | From the Multiple Rule for Continuous Mappings on Metric Space, we have that:
:$\lambda \map f x$ and $\mu \map g x$ are continuous.
From the Sum Rule for Continuous Mappings on Metric Space, we have that:
:$\lambda \map f x + \mu \map g x$ is continuous.
So, by definition of continuous again, we have that $\lambda f +... | :$\lambda f + \mu g$ is [[Definition:Continuous Mapping (Metric Space)|continuous]] on $M$. | From the [[Multiple Rule for Continuous Mappings on Metric Space]], we have that:
:$\lambda \map f x$ and $\mu \map g x$ are [[Definition:Continuous Mapping (Metric Space)|continuous]].
From the [[Sum Rule for Continuous Mappings on Metric Space]], we have that:
:$\lambda \map f x + \mu \map g x$ is [[Definition:Conti... | Combination Theorem for Continuous Mappings/Metric Space/Combined Sum Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Metric_Space/Combined_Sum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Metric_Space/Combined_Sum_Rule | [
"Combination Theorem for Continuous Mappings"
] | [
"Definition:Continuous Mapping (Metric Space)"
] | [
"Combination Theorem for Continuous Mappings/Metric Space/Multiple Rule",
"Definition:Continuous Mapping (Metric Space)",
"Combination Theorem for Continuous Mappings/Metric Space/Sum Rule",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Continu... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.