Let’s prove completeness for the classical Banach spaces, say $L^{p}[0,1]$ where $p\geq 1$.

Since the case $p=\infty$ is elementary, we may assume $1\leq p<\infty$. Let $[f_{{\cdot}}]\in(L^{p})^{{\mathbf{N}}}$ be a Cauchy sequence. Define $[g_{0}]:=[f_{0}]$ and for $n>0$ define $[g_{n}]:=[f_{n}-f_{{n-1}}]$. Then $[\sum_{{n=0}}^{N}g_{n}]=[f_{N}]$ and we see that

Thus it suffices to prove that etc.

It suffices to prove that each absolutely summable series in $L^{p}$ is summable in $L^{p}$ to some element in $L^{p}$.

Let $\{f_{n}\}$ be a sequence in $L^{p}$ with $\sum_{{n=1}}^{\infty}\|f_{n}\|=M<\infty$, and define functions $g_{n}$ by setting $g_{n}(x)=\sum_{{k=1}}^{n}|f_{k}(x)|$. From the Minkowski inequality we have

Hence

For each $x$, $\{g_{n}(x)\}$ is an increasing sequence of (extended) real numbers and so must converge to an extended real number $g(x)$. The function $g$ so defined is measurable, and, since $g_{n}\geq 0$, we have

by Fatou’s Lemma. Hence $g^{p}$ is integrable, and $g(x)$ is finite for almost all $x$.

For each $x$ such that $g(x)$ is finite the series $\sum_{{k=1}}^{\infty}f_{k}(x)$ is an absolutely summable series of real numbers and so must be summable to a real number $s(x)$. If we set $s(x)=0$ for those $x$ where $g(x)=\infty$, we have defined a function $s$ which is the limit almost everywhere of the partial sums $s_{n}=\sum_{{k=1}}^{n}f_{k}$. Hence $s$ is measurable. Since $|s_{n}(x)|\leq g(x)$, we have $|s(x)|\leq g(x)$. Consequently, $s$ is in $L^{p}$ and we have

Since $2^{p}g^{p}$ is integrable and $|s_{n}(x)-s(x)|^{p}$ converges to $0$ for almost all $x$, we have

by the Lebesgue Convergence Theorem. Thus $\|s_{n}-s\|^{p}\to 0$, whence $\|s_{n}-s\|\to 0$. Consequently, the series $\{f_{n}\}$ has in $L^{p}$ the sum $s$.