Properties of a set in $\ell^2$ space

Question

Let $\ell^2 = \{x= (x_1,x_2,x_3,\ldots): x_n\in \mathbb C\text{ and } \sum_{n=1}^\infty |x_n|^2 < \infty\}$

and $e_n \in \ell^2 $ be the sequence whose $n$-th element is $1$ and all other elements are $0$.

Equip the space with $\ell_2$ with the norm $$\|x\| = \left(\sum_{n=1}^\infty |x_n|^2\right)^{1/2}$$

Then the set $S=\{e_n : n \geq 1\}$ is

1. closed;

2. bounded;

3. compact;

4. and the sequence $s=(e_n)_{n\geq 1}$ contains a convergent sub-sequence.

Answer 1

My answer: only 2 is true. It is not closed as $\lim e_n$ does not exist. It is bounded as $||e_n||=1$ It is not compact as one can not find a finite sub-cover containing S. Neither does it contain a convergent subsequence as $\lim e_{n_k} $ does not exist.

Answer 2

Every pair of orthonormal vectors $e_i\neq e_j$ has distance precisely $d(e_i,e_j)=\sqrt{2}$.

So $S$ is closed as it contains only isolated points, it is bounded as all its elements have norm one, it is not compact as it contains infinitely many isolated points, it contains a convergent subsequence the constant ones?!