Prove that the set $A=\{ s\in l^p \mid ||s||_p \leq 1 \}$ is non-compact.

Question

$s=(s_n)_n\in\mathbb{N}$ is a real sequence, and \begin{equation} ||s||_p=\begin{cases} \sup_{n\in\mathbb{N}}\left(\sum_{i=1}^{n}|s_i|^p\right)^{1/p},if ~1\leq p<\infty\\ \sup_{n\in\mathbb{N}}|s_n|, if~ p=\infty \end{cases}\end{equation} $l^p$ is the space of real sequences $s$ with $||s||_p<\infty$. I know maybe we need to construct an open cover that does not have finite subcovers that covers $A$, but do not know how to proceed.

Answer 1

Consider the sequence $\{e_i\}_{i=1}^{\infty}$ where $e_i$ is identically zero except the $i^{\text{th}}$ coordinate. Then no subsequence can converge.

Answer 2

Let $A_n$ be the set of all sequences with arbitrary entries in the first n coordinates and norm of the remaining coordinates smaller than 1/2. This works for all cases except the supremum norm, which I'll leave to you