Proving that a sequence in $\ell^2$ is a Cauchy sequence

Question

Let $(x^{(n)})_{n\in\mathbb{N}}, x^{(n)}:= \sum\limits_{i=1}^n \frac{1}{i} e_i$, where $e_i$ is the sequence that is $0$ everywhere but $1$ in the $i^{th}$ element.

I would like to prove that this sequence is a Cauchy-sequence in a subspace of $\ell^2$.

I have $||x^{(n)}-x^{(m)}||^2 = … = \sum\limits_{i=m+1}^n \frac{1}{i^2}$. This can be of course estimated by $||x^{(n)}-x^{(m)}||^2 \leq \frac{n-m-1}{(m+1)^2}$ or $||x^{(n)}-x^{(m)}||^2 \leq\frac{n-m}{m^2}$ or $||x^{(n)}-x^{(m)}||^2 \leq\frac{n}{m^2}$.

Now I'm wondering if (and how) this helps.

Can someone please give me hint on that?

The Axiom of Archimedes doesn't seem to work here (right?)

Answer 1

Hint: $\sum_{j\geq 1} j^{-2}<\infty$.