unit sphere in $l^\infty$ is not compact

Question

I am trying to solve the following problem:

Let $V=${$v: \mathbb{N} \mapsto \mathbb{R} : ||v|| < \infty$}, where $||v||$= sup$_{n \in \mathbb{N}}|v_n|$

Then the closed unit sphere $B=${$v \in V: ||v||=1 $} is not compact.

How can you show that $B$ is not compact?

Answer 1

This is a metric space, so it is enough to show that there is a sequence (of functions) in $B$ without convergent subsequence. For any $n\in\mathbb{N}$ consider

$$e_n:\mathbb{N}\to\mathbb{R}$$ $$e_n(m)=\begin{cases}1 &\text{if }n= m\\ 0 &\text{otherwise} \end{cases}$$

First of all note that $\lVert e_n\rVert=1$ for any $n$, and so $e_n\in B$.

Now if $n\neq m$ then you can easily check that $\lVert e_n-e_m\rVert=1$. Thus no subsequence of $(e_n)$ is a Cauchy sequence. In particular $(e_n)$ has no convergent subsequence.