Show that a metric space $(X,d)$ is totally bounded(ball compact) if and only if every sequence in X has a Cauchy subsequence.
118328
Metric space is totally bounded iff every sequence has Cauchy subsequence
123061
$X$ metric separable then $C(X)$ separable
108619
Proving Baire's theorem: The intersection of a sequence of dense open subsets of a complete metric space is nonempty
73080
Is the set of all bounded sequences complete?
73469
Continuity of the function $x\mapsto d(x,A)$ on a metric space
130877
Boundedness and Cauchy Sequence: Is a bounded sequence such that $\lim(a_{n+1}-a_n)=0$ necessarily Cauchy?
11586
Sequences of Real Numbers in a closed bounded interval
90628
The metric spaces are Tychonoff spaces
126759
Unbounded continuous function on non-compact metric space
113340
Show that the recursive sequence is bounded
153952
A map between metric spaces preserving convergent sequences is continuous
11114
Is a closed subset of a compact set (which is a subset of a metric space $M$) compact?
133504
Suppose $X$ is a metric space and $S \subseteq X.$ Then, $S^o=\{x \in X~|~dist(x,S^c)>0\}$.
119105
If $(X,d)$ is a complete metric space and $A$ is closed then show that for $x \in X$ there exists an element $a_0 \in A$ such that $d(x,A)=d(x,a_0)$
52014
Showing $(C[0,1], d_1)$ is not a complete metric space
35514
If $\{f_n\}$ converges uniformly and each $f_n$ is bounded, show that $\exists M>0$ s.t. $|f_n(x)| \leq M$
113696
Showing that the unit ball in $(C^\alpha[a, b], \|\cdot\|_\alpha)$ is compact in $(C[a, b], \|\cdot\|_\infty)$