I am trying to prove $L^1$ is complete. Need help with this step:
Consider a sequence $f_k$ in $L^1$ with $$\|{f_{{k+1}}-f_{k}\|}_{L^1} <2^{-k}$$ Define $$g_k(x)=\sum^k_j|f_j(x)-f_{j+1}(x)|$$ show limit $g_k$ exists in $L^1$ and point wise. $\lim_k g_k$ is in $L^1$, since $$\int |\lim_k g_k(x)|\mu(dx)\leq \int \lim_k \sum_j^k |f_{j+1}(x)-f_j(x)|\mu(dx)=\lim_k\int \sum_j^k |f_{j+1}(x)-f_j(x)|\mu(dx)$$ since Beppo Levi $\iff$ finite additivity of the integral and using monotone convergence, so: $$\int \lim_k g_k(x)\mu(dx) \leq \lim_k 2^{-k} <\infty \Rightarrow \lim_kg_k\in L^1.$$
How do I show the point wise limit exists?
Maybe this lemma will clear things up: suppose that $X$ is a measure space and $1 ≤ p < ∞$. If $\{g_k ∈ Lp(X) : k ∈ N\}$ is a sequence of $Lp$-functions such that $\sum \|g_k\|<∞$, then there exists a function $f ∈ L_p(X)$ such that $\sum g_k=f.$
The proof goes as follows:
Define $h_n=\sum^n |g_k|;\ h=\sum^n |g_k|$. Then, $h_n\uparrow h$ (in theory, the sum may be infinite.) In any case, the monotone convergence theorem gives $\int h^p=\lim \int h_n^p.$
On the other hand, Minkowski's inequality says $\|h_n\|_p\le \sum^n\|g_k\|_p\le \sum \|g_k\|_p<\infty$ (the last inequality holds by assumption.) Therefore, $h\in L_p(X).$
So, in fact, $h$ is finite a.e. and this implies that $\sum^n g_k$ converges absolutely a.e. to a function $f\in L_p(X)$ (because $|f|<h\in L_p(X).)$
Now, the dominated convergence theorem applies to show that the sum also converges to $f$ in $L_p(X)$, for $\left | f-\sum^{n} g_k\right |^{p}\le (2h)^{p}$