Existence of a global limit in $L^1([-N,N])$ for each $N\in \mathbb{N}$

Question

Let $(f_n)_n$ be sequence of functions $f_n\in L^1_{loc}(\mathbb{R})$ such that for each $N\in \mathbb{N}$, $(f_n)_n$ is a Cauchy sequence in $L^1([-N,N])$. Then for each $N$, $(f_n)_n$ converges to a function $f_N\in L^1([-N,N])$. Do I need to use something like Zorn's Lemma or axiom of choice to conclude that there exists a function $f\in L^1_{loc}(\mathbb{R})$ such that $(f_n)_n$ converges to $f$ in $L^1$ in each compact interval ? If so, then how to use it ?

Answer 1

You don't need Zorn's lemma. The function $f$ can be defined explicitly as $$ f_1\chi_{\{|x|\le 1\}}+\sum_{n=2}^\infty f_{n} \chi_{\{n-1<|x|\le n\}} $$ which is evidently in $L^1_{\rm loc}$ (on every bounded interval, only finitely many terms are nonzero). The convergence of $f_n$ to $f$ on every bounded interval follows from the fact that the restriction of $f$ to $[-N,N]$ is $$ f_1\chi_{\{|x|\le 1\}}+\sum_{n=2}^N f_{n} \chi_{\{n-1<|x|\le n\}} = f_1+\sum_{n=2}^N f_{n} \chi_{\{|x|\le n\}} - \sum_{n=2}^N f_{n} \chi_{\{|x|\le n-1\}} \\ = f_1\chi_{\{|x|\le 1\}}+\sum_{n=2}^N f_{n} \chi_{\{|x|\le n\}} - \sum_{n=2}^N f_{n-1} \chi_{\{|x|\le n-1\}} = f_N \chi_{|x|\le N} $$