How to verify whether $(C_{00},\|\cdot\|_p)$ is complete

Question

How to verify whether $C_{00}=\{(x_n):\text{All but finitely many terms are }0\}$ is complete with respect to $p$-norm given by $$\|(x_n)\|_p=\left(\sum_{n=1}^\infty|x_n|^p\right)^{1/p}$$ where $1\le p<\infty.$

Answer 1

Calling $e^{(n)}$ the sequence whose all terms are $0$, except the term $n$ with is $1$, and defining $$x^{(n)}:=\sum_{j=1}^n2^{-j}e^{(j)}.$$ It's a Cauchy sequence in $c_{00}$ for each $\lVert\cdot \rVert_p$ norm, but it doesn't converge to an element of this space.

A deeper reason for which $c_{00}$ is not complete for any norm is that it has a countable Hamel basis (namely the collection of $e^{(n)}$). By Baire's theorem, it cannot be complete.