= Solution
First let $1\leq p<\infty$, and take a <Cauchy sequence> $f_j$ in the <Lp space>, with functions identified <almost everywhere>. Choose a subsequence $f_{j_k}$ with $\|f_{j_{k+1}}-f_{j_k}\|_p\leq2^{-k}$ and put $a_k=f_{j_{k+1}}-f_{j_k}$. The <triangle inequality> gives
$$
\left\|\sum_{k=1}^N|a_k|\right\|_p\leq\sum_{k=1}^N2^{-k}\leq1.
$$
By the <monotone convergence theorem>, $A=\sum_{k\geq1}|a_k|$ belongs to $L^p$, and is finite almost everywhere. The series $f_{j_1}+\sum_{k\geq1}a_k$ therefore converges almost everywhere to an $L^p$ function $f$. Applying the same argument to each absolute tail yields
$$
\|f-f_{j_k}\|_p\leq\sum_{\ell\geq k}2^{-\ell}\longrightarrow0.
$$
The original sequence is Cauchy, so the triangle inequality extends convergence from this subsequence to the whole sequence.
For $p=\infty$, choose the same subsequence in <essential supremum> norm. Outside one null set, all its difference bounds hold simultaneously:
$$
|a_k(x)|\leq2^{-k},\qquad |f_{j_1}(x)|\leq\|f_{j_1}\|_\infty.
$$
The series then converges uniformly on that full-measure set to a bounded measurable $f$, and $\|f-f_{j_k}\|_\infty\leq\sum_{\ell\geq k}2^{-\ell}$. Again the full sequence converges. Thus \b[every $L^p$, $1\leq p\leq\infty$, is a Banach space], proving <completeness of Lp spaces>.
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