$\ell_p$ is not closed in $\ell_ \infty $

Question

prove $\ell_p$ is not closed in $\ell _\infty $ for any $1 \le p < \infty$.

A sequence of real numbers if just a function $f:\mathbb N\to\mathbb R$. So, we have a sequence of functions $f_n:\mathbb N\to\mathbb R$ and $f_n \in \ell _p $ for any $n \in \mathbb{N}$. They converge to some function $g$ in the sense that $\|f_n-g\|_p\to 0$. The goal is to prove that the function $g \in \ell _\infty ,g \notin \ell _p $.

Answer 1

Hint: you have to find for each $1 \le p < \infty$ a concrete sequence $f_n$ in $\ell^p (\,\subseteq \ell_\infty)$ (so you have to show that they indeed are in that set, so they're $p$-summable) and $f \in \ell^\infty$ such that $f \notin \ell_p$ (show this!) and such that $\|f_n - f\|_\infty \to 0$ as $n \to \infty$.