$\ell^p(J)$ is complete (Banach)

Question

I have to prove that the space

$\ell^p(J)$ defined as the set of all functions

$\psi: J\rightarrow \mathbb{F}$

s.t. $\psi$ is null except in a contable subset of $J$

and

$||\psi||_p :=\bigg(\sum_{t\in J}|\psi(t)|^p\bigg)^{1/p}<\infty$

is Banach (complete).

Well, given a Cauchy sequence $(\psi_n)_n\subset \ell^p(J)$ I defined $\psi$ such that each $\psi(t)$ is defined as $\lim \psi_n(t)$ (each of these sequences $\psi_n(t)$ is Cauchy in $\mathbb{F}$).

I proved that $\psi_n\rightarrow \psi$ and $||\psi||_p<\infty$. However, I could not prove that $\psi$ is null except in a contable subset of $J$.

Many thanks for any help.

Answer 1

Hint: if $\psi(t) \ne 0$, then some $\psi_n(t) \ne 0$.