Convergence of $\sum_{j=1}^\infty x_j^2$ assuming that the sum of squares is finite.

Question

If I let $l_2$ be the set of all real sequences $\{x_j\}_{j\in N}$, such that

$\sum_{j=1}^{\infty} x_j^2 < \infty$,

is there any way to show that this sum converges? Can I do it by showing that $\{x_j\}_{j\in N}$ is Cauchy?

Answer 1

Let $S_k = \sum_{j=1}^k x_j^2$. By definition the sum $\sum_{j=1}^\infty x_j^2$ converges if and only if $S_k$ converges.

Since $x_i^2>0$, the sequence $S_k$ is monotonically increasing, by assumption it is also bounded and so it converges. In fact, $S_k\rightarrow S$ where $S=\sup \{S_k : k\in\mathbb{N}\}$.

Note that this implies that $x_j^2\rightarrow 0$, hence $x_j\rightarrow 0$. However the fact that $x_j\rightarrow 0$ does not imply that $S_k$ converges.