= Solution
Completeness means that every norm-Cauchy sequence has a limit belonging to the same <Lp space>, with $\|f_n-f\|_p\to0$. To prove it, choose a subsequence $f_{n_j}$ with $\|f_{n_{j+1}}-f_{n_j}\|_p\leq2^{-j}$. The functions
$$
H_N=|f_{n_1}|+\sum_{j=1}^N|f_{n_{j+1}}-f_{n_j}|
$$
have $p$-norm bounded by $\|f_{n_1}\|_p+1$, by <Minkowski inequality>. They increase to $H$, and <Fatou's lemma> gives $\int H^p\leq(\|f_{n_1}\|_p+1)^p$. Thus $H$ is finite almost everywhere, so the subsequence telescopes absolutely there to a measurable limit $f$ with $|f|\leq H$ and $f\in L^p$.
The same argument on the tails gives $\|f-f_{n_j}\|_p\leq\sum_{k\geq j}2^{-k}\to0$. Given $\epsilon$, choose such a subsequence term beyond the Cauchy threshold and use $\|f_n-f\|_p\leq\|f_n-f_{n_j}\|_p+\|f_{n_j}-f\|_p$. This proves convergence of the full sequence and \b[completeness for every $1\leq p<\infty$], on an arbitrary measure space.
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