Absolute convergence of $\sum_{n=0}^{\infty}a_n\implies\sum_{n=0}^{\infty}a_n^2$ is absolute convergent too I am pretty sure that

Absolute convergence of $\sum_{n=0}^{\infty}a_n\implies\sum_{n=0}^{\infty}a_n^2$ is absolute convergent too

is a true statement, but before I proof it I ask wether this exercise is probably meant for series in $\mathbb{R}$ respectivly $\mathbb{C}$ or is it meant for ANY series. Like series of vectors in an unknown vectorspace.
Could someone clarify me?
I don't know what I mumble so here is what I stumbled on, while searching for definitions of absolute convergence: https://proofwiki.org/wiki/Definition:Absolute_Convergence/General

Since $\sum_n a_n$ converges absolutely, only finitely many $|a_n| > 1$. Hence, except for these finitely many terms, $|a_n^2| = |a_n|^2 \le |a_n|$, and thus $\sum_n a_n^2$ converges absolutely. Note this solution works over $\mathbb{R}$, $\mathbb{C}$, and I guess any normed algebra.
