Convergence of sequences in metric spaces

Show that each sequence in a metric space has at most one limit.

Prove that in the space $B_{0}$ the limit of the sequence $\left\{ x_{n} = \left(x_{n}^{1}, x_{n}^{2}, \ldots \right)\right\}_{1}^{\infty }$ equals $x = \left(x^{1}, x^{2}, \ldots \right)$ if and only if for every $k \in \mathbb {N}$ there exists $N_{k} \in \mathbb {N}$ such that $x_{n}^{k} = x^{k}$ for all $n > N_{k}$.

Prove that in a metric space every convergent sequence is a Cauchy sequence.

Prove that if a sequence $\left\{ x_{n}\right\}_{n=1}^{\infty }$ does not converge to $x$ then there is an open ball $B(x, r)$ that contains no members of the sequence.

Let $G$ be an open set in $\mathbb {R}^{n}$, let $x_{0} \in \mathbb {R}^{n}$, and let $\left\{ x_{n}\right\}$ be a sequence in $\mathbb {R}^{n}$ that converges to $x_{0}$. Prove that $G+x_{0} \subset \liminf \left(G+x_{n}\right)$.

Let $F$ be a closed set in $\mathbb {R}^{n}$, let $x_{0} \in \mathbb {R}^{n}$, and let $\left\{ x_{n}\right\}$ be a sequence in $\mathbb {R}^{n}$ that converges to $x_{0}$. Prove that $F+x_{0} \supset \limsup \left(F+x_{n}\right)$.

Let $A$ be a set in a metric space $M$ and let $a$ be a cluster point of $A$. Prove that there is a sequence $\left\{ a_{n}\right\} \subset A$ that converges to $a$.

Let ( $M, d$ ) be a metric space, and let $\lim x_{n}=x, \lim y_{n}=y$. Prove that $\lim d\left(x_{n}, y_{n}\right)=d(x, y)$.

Let $\left\{ x_{n}\right\}$ be a convergent sequence in a metric space. Prove that every subsequence of $\left\{ x_{n}\right\}$ converges to the same limit.

Let $\left\{ x_{n}\right\}$ be a sequence in a metric space and suppose that each of its three subsequences $\left\{ x_{2 n}\right\} ,\left\{ x_{2 n+1}\right\}$, and $\left\{ x_{3 n}\right\}$ is convergent. Prove that $\left\{ x_{n}\right\}$ is a convergent sequence.

Let $\left\{ x_{n}\right\}$ be a sequence in a metric space and suppose that each of its infinitely many subsequences $\left\{ x_{s n}\right\}$, where $s=2,3, \ldots$, is convergent. Prove or disprove: $\left\{ x_{n}\right\}$ is a convergent sequence.

Let $\left\{ x_{n}\right\}$ be a sequence in a metric space ( $M, d$ ), let $x \in M$, and suppose that each subsequence $\left\{ x_{n_{k}}\right\} \subset \left\{ x_{n}\right\}$ has a further subsequence $\left\{ y_{n}\right\}$ which converges to $x$. Prove that $\left\{ x_{n}\right\}$ converges to $x$.

Let ( $X, d$ ) be a metric space that is not complete and let $\left\{ x_{n}\right\}$ be a Cauchy sequence in $X$ that is not convergent. Prove that the set $E=\left\{ x_{n}\right\}$ is closed.

Let $(X, \rho )$ be a metric space in which $\left\{ u_{n}\right\} \rightarrow u$ and $\left\{ v_{n}\right\} \rightarrow v$. Show that $\left\{ \rho \left(u_{n}, v_{n}\right)\right\} \rightarrow \rho (u, v)$.

If $s_{k} = \left\{ x_{k n}\right\}_{n \geq 1}$ and $s = \left\{ x_{n}\right\}_{n \geq 1}$, are elements of $\mathbb {S} = \mathbb {R}^{\infty }$ as in Problem A.26, verify that as $k \rightarrow \infty , s_{k} \rightarrow s$ iff $x_{k n} \rightarrow x_{n}$ for all $n \geq 1$.

Establish Theorem A.4.1.