I was looking at the proof that $l^{p}$ is complete with respect to the standard metric.
Suppose $x^{(n)}$ is a Cauchy sequence in $l^{p}$. Then
Given $\epsilon > 0$, $\exists\,\, n_{0} \in \mathbb{N}$ such that
$$ \sum\limits_{j=1}^{\infty} |x^{(n)}_{j} j- x^{(m)}_{j}|^{p} < \epsilon^{p}$$ for all $m,n \geq n_{0}$ Thus $$ \sum\limits_{j=1}^{N} |x^{(n)}_{j} j- x^{(m)}_{j}|^{p} < \epsilon^{p}$$ for all $N \in \mathbb{N}$ provided $m,n \geq n_{0}$.
First of this implies that $x^{(n)}_{j}$ converges for every fixed $j$ to some real number $x_{j}$. Then they take the limit $m \rightarrow \infty$ in the second equation and conclude (for $n$ fixed)
$$ \sum\limits_{j=1}^{N} |x^{(n)}_{j} j- x_{j}|^{p} < \epsilon^{p}.$$
This is where my problem is. Could someone explain how this taking limits is justified?
Note that for fixed $j \in \mathbb{N}$, we have
$$|x_j^{(n)}-x_j^{(m)}| \leq \left( \sum_{j=1}^{\infty} |x_j^{(n)}-x_j^{(m)}|^p \right)^{\frac{1}{p}} < \varepsilon$$
i.e. $(x_j^{(n)})_{n \in \mathbb{N}}$ is a Cauchy sequence in $\mathbb{R}$, thus $x_j^{(n)} \to x_j$ as $n \to \infty$ for suitable $x_j \in \mathbb{R}$. Consequently, for fixed $N \in \mathbb{N}$ (so it's a finite sum!),
$$\sum_{j=1}^N |x_j^{(n)}-x_j^{(m)}|^p \stackrel{m \to \infty}{\to} \sum_{j=1}^N |x_j^{(n)}-x_j|^p < \varepsilon^p$$
thus,
$$\sum_{j=1}^N |x_j^{(n)}-x_j|^p \leq \varepsilon^p.$$