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

*

*closed;


*bounded;


*compact;


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

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?!
