compactness in $\ell^p$ space

Question

Choose $1 \leq p \leq \infty$, and let $D=\left\{x \in \ell^{p}:\|x\|_{p} \leq 1\right\}$ be a closed ball in $\ell^p$. Try to show that $D \text { is not a compact subset of } \ell^{p}$.

So far I've proved that the sequence of standard basis vectors $\left\{\delta_{n}\right\}_{n \in \mathbb{N}}$ contains no convergent subsequences, will that directly imply $D$ is not compact?

Any help is appreciated.

Answer 1

Firstly, using F. Riesz's lemma, you can prove that the closed unit ball is compact in a normed space if and only if the respective normed space is finite-dimensional, which $\ell^p$ is not.

Secondly, in a metric space, compactness is equivalent with sequential compactness, so yes, if you find a sequence that has no convergent subsequence, then your space is not compact.