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If $f$ is a continuous function on an open set $S$, and $f(x) \neq a$ for some $x \in S$, then there exists an $\epsilon > 0$ such that $f(y) \neq a$ for all $y$ with $|x - y| < \epsilon$. |
If $S$ is a nonempty set of real numbers that is bounded below and closed, then $\inf S \in S$. |
If $C$ is a closed set and $A$ is a nonempty subset of $C$ that is bounded below, then $\inf A \in C$. |
If $S$ is a nonempty set of real numbers that is bounded above and closed, then $\sup S \in S$. |
If $C$ is a closed set, $A$ is a nonempty bounded above subset of $C$, then $\sup A \in C$. |
If $A$ is a nonempty subset of the closed interval $[a,b]$, then $\inf A \in [a,b]$. |
A set of real numbers is bounded if and only if there exists a real number $a$ such that for all $x$ in the set, $|x| \leq a$. |
If $S$ is a bounded set of real numbers, then $S$ is bounded above. |
If $S$ is a bounded set of real numbers, then $S$ is bounded below. |
If $S$ is a nonempty bounded set of real numbers, then $\sup S$ exists and is an upper bound for $S$. |
If $S$ is a bounded set, then $\sup(S \cup \{x\}) = \max\{x, \sup S\}$. |
If $S$ is a nonempty bounded set of real numbers, then $\inf S$ is the greatest lower bound of $S$. |
If $S$ is a bounded set, then $\inf(S \cup \{x\}) = \min\{x, \inf S\}$. |
A subset of the real numbers is open if and only if for every point in the set, there is an open interval around that point that is contained in the set. |
A real number $x$ is a limit point of a set $S$ if and only if for every $\epsilon > 0$, there exists a point $x' \in S$ such that $x' \neq x$ and $|x' - x| < \epsilon$. |
A set $S$ is closed if and only if for every $x$, if for every $\epsilon > 0$, there exists $x' \in S$ such that $x' \neq x$ and $|x' - x| < \epsilon$, then $x \in S$. |
A function $f$ is continuous at $x$ if and only if for every $\epsilon > 0$, there exists a $\delta > 0$ such that for all $x'$, if $|x' - x| < \delta$, then $|f(x') - f(x)| < \epsilon$. |
A function $f$ is continuous on a set $S$ if and only if for every $x \in S$ and every $\epsilon > 0$, there exists a $\delta > 0$ such that for all $x' \in S$, if $|x' - x| < \delta$, then $|f(x') - f(x)| < \epsilon$. |
If $f$ and $g$ are continuous real-valued functions defined on a closed set $S$, then the set $\{x \in S \mid f(x) \leq g(x)\}$ is closed. |
If $f$ is continuous on the closure of $S$ and $f(x) \leq a$ for all $x \in S$, then $f(x) \leq a$ for all $x \in \overline{S}$. |
If $f$ is continuous on the closure of $s$ and $f(x) \geq a$ for all $x \in s$, then $f(x) \geq a$ for all $x \in \overline{s}$. |
The distance between the empty set and any set is zero. |
The distance between a set and the empty set is zero. |
The distance between two sets is nonnegative. |
If $s$ and $t$ are nonempty sets and $d$ is a real number such that for all $x \in s$ and $y \in t$, we have $d \leq \text{dist}(x, y)$, then $d \leq \text{setdist}(s, t)$. |
If $x$ is in $s$ and $y$ is in $t$, then the distance between $s$ and $t$ is less than or equal to the distance between $x$ and $y$. |
The distance between two sets $S$ and $T$ is the infimum of the distances between points in $S$ and $T$. |
If the distance between two nonempty sets is less than $b$, then there exist points $x \in S$ and $y \in T$ such that the distance between $x$ and $y$ is less than $b$. |
The distance between a set and itself is zero. |
The distance between two sets is symmetric. |
The distance between two sets is less than or equal to the sum of the distances between the first set and a point and the point and the second set. |
The distance between two sets is the distance between their elements. |
The distance between a point and a set is Lipschitz. |
The function $y \mapsto \text{dist}(\{y\}, S)$ is continuous at $x$. |
The function $y \mapsto \text{dist}(\{y\}, S)$ is continuous on $T$. |
The function $y \mapsto \operatorname{dist}(\{y\}, S)$ is uniformly continuous on $T$. |
If $T$ is a nonempty subset of $u$, then the distance from $S$ to $u$ is less than or equal to the distance from $S$ to $T$. |
If $S$ is a nonempty subset of $T$, then the distance from $u$ to $T$ is less than or equal to the distance from $u$ to $S$. |
The distance between a set and its closure is zero. |
The distance between a set $T$ and the closure of a set $S$ is the same as the distance between $T$ and $S$. |
If $x$ is in both $S$ and $T$, then the distance between $S$ and $T$ is zero. |
If $a \in S$ and $b \<in> T$ are such that $dist(a,b) \leq dist(x,y)$ for all $x \in S$ and $y \in T$, then $setdist(S,T) = dist(a,b)$. |
If $x \in S$, then the distance between $S$ and $T$ is less than or equal to the distance between $\{x\}$ and $T$. |
The infimum of the distance from $x$ to $A$ is equal to the distance from the set $\{x\}$ to $A$. |
The distance between two sets is equal to the infimum of the distances between the points of the first set and the second set. |
If $A \subseteq B$ and $A$ is nonempty, then the infimum of the distances from $x$ to $B$ is less than or equal to the infimum of the distances from $x$ to $A$. |
The infimum distance from a point $x$ to a singleton set $\{y\}$ is the distance from $x$ to $y$. |
If $B$ is a nonempty compact set, then there exists a point $y \in B$ such that the distance from $A$ to $B$ is equal to the distance from $y$ to $A$. |
If $S$ or $T$ is open, then the Minkowski sum $S + T$ is open. |
If $f$ is an orthogonal transformation, then $f$ maps the ball of radius $r$ centered at $x$ to the ball of radius $r$ centered at $f(x)$. |
If $f$ is an orthogonal transformation, then $f$ maps the closed ball of radius $r$ centered at $x$ to the closed ball of radius $r$ centered at $f(x)$. |
$x$ is in the support of $f$ on $S$ if and only if $x$ is in $S$ and $f(x) \neq 0$. |
The support of a function $f$ on a set $S$ is the set of all elements of $S$ where $f$ is nonzero. |
If two functions agree on their support, then they have the same support. |
If $a \neq 0$, then the support of the function $x \mapsto a$ if $P(x)$ holds and $0$ otherwise is the set of all $x$ such that $P(x)$ holds. |
If $A$ is a set and $P$ is a predicate on $A$, then the support of the function $f(x) = a$ if $P(x)$ and $f(x) = 0$ otherwise is a subset of the set of all $x \in A$ such that $P(x)$ holds. |
If $S$ is a finite set, then the support of a function $f$ on $S$ is finite. |
The support of the sum of the empty set is zero. |
If $f$ is a function with finite support on $S$, then the support of $f$ on $S \cup \{x\}$ is equal to the support of $f$ on $S$ if $x \in S$, and is equal to $f(x) +$ the support of $f$ on $S$ if $x \notin S$. |
The support of a sum divided by a scalar is the support of the sum of the scalar divided by the scalar. |
The image of an interval under an affine transformation is an interval. |
A point $y$ is a limit point of the open ball $B(x,\epsilon)$ if and only if $0 < \epsilon$ and $y \in \overline{B(x,\epsilon)}$. |
If $x \neq y$, then $y$ is a limit point of the ball of radius $|x - y|$ centered at $x$. |
For any vector $x$ and any real number $e$, $x \in B(0, e)$ if and only if $\|x\| < e$. |
For any vector $x$ and any real number $e$, $x \in B(0, e)$ if and only if $\|x\| \leq e$. |
The closure of a ball is the ball. |
The set of points on the sphere of radius $e$ centered at the origin is the set of points with norm $e$. |
The interior of a closed ball is the open ball of the same radius. |
The boundary of a ball is the sphere. |
The boundary of a closed ball is the sphere of the same radius. |
The sphere of radius $r$ around $a$ is compact. |
The sphere of radius $r$ around a point $a$ is bounded. |
The sphere of radius $r$ around $a$ is closed. |
The image of a ball under addition is a ball. |
The image of a closed ball under addition is a closed ball. |
The image of the natural numbers under the function $x \mapsto x$ is a closed set. |
The image of the integers under the function that maps each integer to its real counterpart is a closed set. |
The set of natural numbers is closed. |
The set of integers is closed. |
If $A$ is a subset of the integers, then $A$ is closed. |
The limit at infinity is not trivial. |
The filter at $x$ within the ball of radius $r$ centered at $y$ is the empty filter if and only if $r=0$ or $x$ is not in the ball. |
A function $f$ converges to $l$ at infinity if and only if for every $\epsilon > 0$, there exists a $b$ such that for all $x$ with $|x| \geq b$, we have $|f(x) - l| < \epsilon$. |
If $f$ is a function such that for every $\epsilon > 0$, there exists a real number $B$ such that for all $x$ with $|x| \geq B$, we have $|f(x) - l| \leq \epsilon$, then $f$ converges to $l$ at infinity. |
If the sets $S$ and $T$ are eventually equal in a neighborhood of $a$, then the limits of $f$ at $a$ within $S$ and $T$ are equal. |
A net $f$ converges to $l$ if and only if the net $f - l$ converges to $0$. |
If $f$ is eventually bounded by $g$ and $g$ converges to $0$, then $f$ converges to $0$. |
If $f_n$ and $g_n$ are sequences of vectors such that $||f_n|| \leq ||g_n||$ for all $n$ and $g_n \rightarrow 0$, then $f_n \rightarrow 0$. |
If $f$ converges to $0$ and $g$ is bounded, then $f \cdot g$ converges to $0$. |
If $f$ converges to $0$ and $g$ is bounded, then $gf$ converges to $0$. |
If $f$ converges to $0$ and $g$ is bounded, then $f \cdot g$ converges to $0$. |
If $f$ is a function from a topological space to a normed vector space, and if $f$ converges to $l$, then the norm of $l$ is bounded by the norm of $f$. |
If $f$ is a function from a topological space to a normed vector space, and if $f$ converges to $l$, and if $f$ is eventually bounded below by $e$, then $e \<le> \|l\|$. |
If $f$ and $g$ converge to $l$ and $m$, respectively, and $h$ is a bounded bilinear function, then $h(f, g)$ converges to $h(l, m)$. |
If $f$ converges to $l$ at $a$, then $f(a + x)$ converges to $l$ at $0$. |
The limit of a net at a point is the point itself. |
If $f$ is a continuous function on the closure of a set $S$ and $f$ is bounded on $S$, then $f$ is bounded on the closure of $S$. |
A set $S$ is bounded if and only if there exists a positive real number $b$ such that for all $x \in S$, we have $|x| \leq b$. |
A set $S$ is bounded if and only if there exists a positive real number $b$ such that for all $x \in S$, we have $|x| < b$. |
A sequence $f$ is bounded if and only if the range of $f$ is bounded. |
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