Convergent subsequence in a bounded sequence of a complete metric space

Question

Consider a complete metric space E with the following property:

If $x_n$ is a bounded sequence, then $\forall \epsilon > 0$, $\exists i,j , i \neq j$ such that $d(x_i,x_j) < \epsilon$. Question:
Given any such bounded sequence in $E$, can we always find a convergent subsequence ? I am unable to construct a convergent subsequence for an arbitrary bounded sequence in E - the given property seems too weak! Am I missing something?

Answer 1

Let $(x_n)_{n=1}^\infty$ be a bounded sequence in $E$. We claim that $X := \{x_n\}_{n=1}^\infty$ is totally bounded. Indeed, suppose this were not the case. Then there is some $\epsilon >0$ with the following property:

For every finite set $F \subseteq X,$ one can find $x \in X$ such that $d(x,y) \geqq \epsilon$ whenever $y \in F$.

We construct a subsequence $(x_{n_k})_{k=1}^\infty$ as follows. Put $n_1 :=1$. Suppose $n_1<...< n_{k-1}$ were obtained so that $d(x_{n_i},x_{n_j})\geqq \epsilon$ whenever $1\leqq i,j <k$, $i\neq j$. Since we assumed $X$ is not totally bounded, the balls

$$B(x_{n_1};\epsilon),...,B(x_{n_{k-1}};\epsilon)$$

must fail to cover $X$, and there must be an infinite amount of points of $X$ outside their union. Let $n_k>n_{k-1}$ be so that

$$x_{n_k} \not\in B(x_{n_1};\epsilon)\cup...\cup B(x_{n_{k-1}};\epsilon).$$

Thus, the bounded sequence $(x_{n_k})_{k=1}^\infty$ is such that $d(x_{n_i},x_{n_j})\geqq \epsilon$ whenever $i \neq j$, and the stated property of $E$ fails.

It follows that if $E$ has the property, $X$ must be totally bounded. Since $E$ is complete, $X$ must be relatively compact (its closure must be compact). Hence, the sequence $(x_n)_{n=1}^\infty$ in $X$ has a convergent subsequence.

Answer 2

Here is an approach to solving: Suppose $E$ has that property.

First prove: For any bounded sequence $\{x_k\}_{k=1}^{\infty}$ in $E$ and for any $\epsilon>0$, there exists a point $x_n$ that is within a distance $\epsilon$ of infinitely many other points in the sequence.

Next: Use this property to construct a Cauchy sequence.