TITLE: Completeness of $\ell^2$ done right
QUESTION [6 upvotes]: One of our problem sets this quarter in analysis asked us to show that $\ell^2$, the set of all square-summable sequences, is complete. At the time, I struggled to prove this, but I have an idea after reading on in Rudin's Principles. My idea is to use uniform convergence of a sequence of continuous functions and change the order of limits. Please let me know what I need to make the following go through; I have seen many questions here on StackExchange, similar to this, where the answerer skirts the question of this exchange of limits, and I would like if someone could just answer with as many epsilons as needed to get the thing settled once and for all.
We have shown that $\ell^2$ is an inner-product space, and it has a norm $\lVert \mathbf x\rVert^2 = \sum_{n\ge 1} x_n^2$. One step of the process is to come up with a candidate limit for a Cauchy sequence, and we can do this fairly easily. Let $(\mathbf x_n)$ be a Cauchy sequence in $\ell^2$ with $\mathbf x_n = x_{n,1},x_{n,2},x_{n,3},\dotsc$. We can arrange the sequences $\mathbf x_n$ into an infinite square matrix $(x_{i,j})$, and show without much trouble that for a fixed $j\in \mathbf N$ the sequence $(x_{i,j})_{i\in\mathbf N}$ is Cauchy (the columns of the infinite array are themselves Cauchy).
For each fixed $j\in \mathbf N$, and with $m,n$ sufficiently large, $|x_{n,j}-x_{m,j}|\le\lVert \mathbf x_n-\mathbf x_m\rVert < \varepsilon$, since $(\mathbf x_n)$ is Cauchy. Since $\mathbf R$ is complete, put $y_j = \lim_{n\to\infty}x_{n,j}$. We claim that $\mathbf y = y_1,y_2,y_3,\dotsc$ is in $\ell^2$ and $\mathbf x_n\to \mathbf y$. By the reverse-triangle inequality, for sufficiently large $m,n\in\mathbf N$, $|\lVert \mathbf x_n\rVert-\lVert\mathbf x_m\rVert| \le \lVert \mathbf x_n-\mathbf x_m\rVert < 1$ by the Cauchy criterion, so that since $\lVert \mathbf x_m\rVert = C < \infty$, we have $\lim_{n\to\infty}\lVert \mathbf x_n\rVert \le C + 1 < \infty$.
We want to show that $\sum_{j=1}^\infty y_j^2 < \infty$, and here is how I want to do that. For each $n=1,2,\dotsc$, put $f_n:\mathbf N\to \mathbf R$ to be the function $f_n(k) = \sum_{j=1}^kx_{n,j}^2$. Every function from $\mathbf N$ to $\mathbf R$ is continuous, so $(f_n)$ is a sequence of continuous functions. Now, $\sum_{j=1}^\infty y_j^2 = \lim_{k\to\infty}\lim_{n\to\infty}\sum_{j=1}^kx_{n,j}^2 = \lim_{k\to\infty}\lim_{n\to\infty}f_n(k)$. If only I could interchange these limits, then I would have $\sum_{j=1}^\infty y_j^2 = \lim_{n\to\infty}\sum_{j=1}^\infty x_{n,j}^2 = \lim_{n\to\infty}\lVert \mathbf x_n\rVert^2 < \infty$, as we already saw. My thoughts are that, as $k\to\infty$, $f_n(k)\to \lVert \mathbf x_n\rVert^2$, and as $n\to\infty, f_n(k)\to\sum_{j=1}^ky_j^2$, and that this latter convergence is uniform (we are dealing with a finite sum, so the uniform convergence shouldn't be hard to demonstrate is my gut reaction). Thus, if we put $f(k) = \sum_{j=1}^k y_j^2$, then $f_n\to f$ uniformly on $\mathbf N$, so we can exchange the order of the limits, just like in Rudin's theorem 7.11.
I am not sure if this is correct—particularly the part where I am asserting $f_n\to f$ uniformly on $\mathbf N$. If anyone could help straighten this out, I'd be much obliged.
Edit I am aware that this isn't all that has to be said to justify completeness. My main question is the rigorous justification of switching limits.

REPLY [2 votes]: Your prerequisites that $\|\mathbf x_n\|$ is bounded and converges to $y$ pointwise is not enough for uniform convergence. In fact, it gives only weak convergence. A simple counterexample is $\mathbf x_n$ with $x_{n,j}=0$ for $j\ne n$ and $x_{n,n}=1$. 
You really need to make more use of the fact that $\mathbf x_n$ is Cauchy. Denote by $a^{[k]}$ the truncation of the sequence $a$
$$
a^{[k]}=\{a_1,a_2,a_3,\ldots,a_k,0,0,\ldots\}.
$$
Then for $m,n\ge N$ we have for every fixed $k$
$$
\left|\|\mathbf x_n^{[k]}\|-\|\mathbf x_m^{[k]}\|\right|\le\|\mathbf x_n^{[k]}-\mathbf x_m^{[k]}\|\le\|\mathbf x_n-\mathbf x_m\|\le\epsilon.
$$
Taking the limit for $m\to+\infty$ we get for $n\ge N$
$$
\left|\|\mathbf x_n^{[k]}\|-\|\mathbf y^{[k]}\|\right|\le\epsilon,\ \forall k,
$$
that is $\|\mathbf x_n^{[k]}\|$ converges to $\|\mathbf y^{[k]}\|$ uniformly.
This is all you need if you exchange the limits for
$$
\phi_n(k)=\|\mathbf x_n^{[k]}\|=\sqrt{f_n(k)},\qquad \phi(k)=\|\mathbf y^{[k]}\|=\sqrt{f(k)}.
$$
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