R^ω Under the l^2 Metric is Complete (7.43.7)

James Munkres Topology, chapter 7.43, exercise 7:

Show that the subspace of $ℝ^ω$ of those sequences $(x_n)$ such that $\sum x_n^2$ converges is complete under the $\ell^2$ metric.

Proof: Let $(f_n)$ be a Cauchy sequence under this metric. Since the $\ell^2$ distance between any two points is at least as large as the uniform distance in $ℝ^ω$, and since the metric under the latter is complete is complete, let $f_n→f$ in the uniform topology. It suffices to show $f_n→f$ in the $\ell^2$ metric. Let $ε > 0$; let $N$ be such that $d_{\ell^2}(f_n,f_m) < ε/2$ for $n,m≥N$. We shall proceed by showing $d_{\ell^2}(f,f_N) ≤ ε/2$ so that $d_{\ell^2}(f,f_n) < ε$ for all $n≥N$, and this former will be demonstrated by showing $d_{\ell^2}(f,f_N) > ε/2$ implies a neighborhood about $f$ in the uniform topology that does not intersect any $f_n$ for $n≥N$, a contradiction.

Therefore, assume $d_{\ell^2}(f,f_N) > ε/2$, and let $$\sqrt{(f(1)-f_N(1))^2+...+(f(n)-f_N(n))^2} = ε/2+δ$$ for some $n$ and $δ > 0$. Consider the uniform $δ/\sqrt{n}$ neighborhood $U$ about $f$; for any $g∈U$, it is evident that $d_{\ell^2}(f_N,g) ≥ ε/2$ (otherwise, consider the first $n$ coordinates of $f$, $f_N$, and $g$ in $ℝ^n$ and apply the triangle inequality), so that $g≠f_m$ for any $m≥N$.$~\square$