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 $1 \le p < \infty$ . 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 $$\sum_{n=0}^\infty \|g_n\| = \sum_{n=0}^\infty \|f_n - f_{n-1}\| \leq ??? < \infty.$$ 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 $$ \|g_n\|\le\sum_{k=1}^n\|f_k\|\le M. $$ Hence $$ \int g_n^p\le M^p. $$ 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 $g_n\ge 0$ , we have $$ \int g^p\le M^p $$ 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 $|s_n(x)|\le g(x)$ , we have $|s(x)|\le g(x)$ . Consequently, $s$ is in $L^p$ and we have $$ |s_n(x)-s(x)|^p\le 2^p\,[g(x)]^p. $$ Since $2^pg^p$ is integrable and $|s_n(x)-s(x)|^p$ converges to $0$ for almost all $x$ , we have $$ \int|s_n-s|^p\to 0 $$ 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 $s$ .

Bibliography

Royden, H. L. Real analysis. Third edition. Macmillan Publishing Company, New York, 1988.