= Solution
Let $(f_n)$ be a <Cauchy sequence> in the <L2 space> $L^2(\mathbb R)$, choosing measurable representatives. There is a subsequence $(f_{n_k})$ with
$$
\|f_{n_{k+1}}-f_{n_k}\|_2\leq2^{-k}\qquad(k\geq1).
$$
Write $d_k=f_{n_{k+1}}-f_{n_k}$ and $S_N=\sum_{k=1}^N|d_k|$. Repeated <Minkowski inequality> gives $\|S_N\|_2\leq\sum_{k=1}^N2^{-k}\leq1$. As $S_N^2$ increases pointwise, the <monotone convergence theorem> gives
$$
\int_{\mathbb R}\left(\sum_{k=1}^{\infty}|d_k|\right)^2dm
=\lim_{N\to\infty}\int_{\mathbb R}S_N^2dm\leq1.
$$
Hence $S=\sum_k|d_k|$ is finite <almost everywhere> and represents an <L2 function>. Define
$$
f=f_{n_1}+\sum_{k=1}^{\infty}d_k
$$
where the series converges, and set $f=0$ on the exceptional <Lebesgue null set>. It is a <measurable function>, and $|f|\leq|f_{n_1}|+S$ <almost everywhere>, so <Minkowski inequality> gives $f\in L^2(\mathbb R)$.
For fixed $k$, <Fatou's lemma> applied to squares of finite tails and then <Minkowski inequality> give
$$
\|f-f_{n_k}\|_2^2
\leq\liminf_{M\to\infty}\left\|\sum_{j=k}^{M}d_j\right\|_2^2
\leq\left(\sum_{j=k}^{\infty}2^{-j}\right)^2=2^{2-2k}.
$$
Thus the subsequence converges to $f$ in <norm>. Given $\varepsilon>0$, choose a sufficiently late $n_k$ with $\|f-f_{n_k}\|_2<\varepsilon/2$ and with $\|f_n-f_{n_k}\|_2<\varepsilon/2$ for every sufficiently large $n$, using the original <Cauchy sequence> property. The <triangle inequality> then gives $\|f_n-f\|_2<\varepsilon$. Therefore
$$
\boxed{L^2(\mathbb R)\text{ is complete, hence a Banach space}.}
$$
This proves the needed instance of <completeness of Lp spaces> without assuming the result being proved.
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