proof that $L^p$ spaces are complete

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

Since the case $p=\mathrm{\infty }$ is elementary, we may assume .Let $[f_{\cdot }]\in {(L^p)}^{𝐍}$ 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 $L^p$ issummable in $L^p$ to some element in $L^p$.

Let $\{f_n\}$ be a sequence in $L^p$ with, and define functions $g_n$ bysetting $g_n(x)={\sum }_{k=1}^n|f_k(x)|$. From the Minkowski inequality wehave

$$\parallel g_n\parallel \le \sum _{k=1}^n\parallel f_k\parallel \le M.$$

Hence

$$\int g_n^p\le M^p.$$

For each $x$, $\{g_n(x)\}$ is an increasing sequence of (extended) realnumbers and so must converge to an extended real number $g(x)$. Thefunction $g$ 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 foralmost all $x$.

For each $x$ such that $g(x)$ is finite the series ${\sum }_{k=1}^{\mathrm{\infty }}{f}_k(x)$is an absolutely summable series of real numbers and so must be summableto a real number $s(x)$. If we set $s(x)=0$ for those $x$ where$g(x)=\mathrm{\infty }$, we have defined a function $s$ which is the limit almosteverywhere of the partial sums $s_n={\sum }_{k=1}^n{f}_k$. Hence $s$ ismeasurable. 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 $2^p{g}^p$ is integrable and ${|s_n(x)-s(x)|}^p$ converges to $0$ foralmost all $x$, we have

$$\int {|s_n-s|}^p\to 0$$

by the Lebesgue Convergence Theorem. Thus ${\parallel s_n-s\parallel }^p\to 0$, whence$\parallel s_n-s\parallel \to 0$. Consequently, the series $\{f_n\}$ has in $L^p$ the sum$s$.