Every convergent sequence is relatively compact

Question

Let $X$ be an Hilbert space. It is known that:

1. Every bounded set is relatively compact (i.e., has compact closure) iff $X$ is finite dimensional

2. Every convergent sequence is bounded.

3. Every totally bounded set is relatively compact, see here.

My question is: Does there exist an infinite dimensional $X$ such that every convergent sequence is relatively compact?

Answer 1

In any metric space, if $(x_n)_{n\in\mathbb N}$ is a convergent sequence and if $x=\lim_{n\to\infty}x_n$, then $\{x_n\,|\,n\in\mathbb{N}\}$ is relatively compact, since its closure is $\{x_n\,|\,n\in\mathbb{N}\}\cup\{x\}$, which is compact.