This is my homework question and I am not sure if my solution is correct.
First, definition of equisummable:
Let $S \subset \ell^2$. The subset $S$ is called equisummable if for any $\epsilon>0$ there exists $N \in \mathbb{N}$ such that $\forall x \in S$ we have $\sum_{i=N}^\infty \, |x_i| ^{2}<\epsilon$.
And the question is
Let $a\in \ell^2$, $S= \{ x \in \ell^2 \,:\, |x_n|\leq |a_n| \text{ for all } n \in \mathbb{N} \}$. Is $S$ equisummable or not?
My answer is yes, it is equisummable.
Otherwise, $\exists \epsilon>0$ such that $\forall N\in \mathbb{N}$ there exists $x \in S$ (depends on $N$) such that $\sum \limits_{i=N}^\infty \, |x_i|^2 \gt \epsilon$. Since $a_n$ is equal or greater than $x_n$ for all $x \in S$, $\forall N\in \mathbb{N}$ we have $\sum_{i=N}^\infty \, |a_i|^2 \gt \epsilon$. This contradicts with the fact that $\sum \limits_{i=1}^\infty \, |a_i|^2 \lt \infty$ (since $a\in \ell^2$).
Is my answer correct? And do I need to explain the last part a little bit more (the reason they contradict)? It seems obvious to me but I am not sure.
Thank you in advance.