I am working on the following problem:
Let $(f_n)$ be a sequence of measurable real-valued functions on $\mathbb{R}$. Prove that there exist constants $c_n > 0$ such that the series $\sum c_n f_n$ converges for almost every $x$ in $\mathbb{R}$. (Hint: Use Borel-Cantelli Lemma).
I am thinking to use an approach similar to the one in Proposition 2 here http://www.austinmohr.com/Work_files/prob2/hw1.pdf , where we pick $c_n$ such that $\mu({x: c_n f_n(x) > 1}) < 1/2^n$.
But I don't understand if and why there exists such a $c_n$, i.e. consider an unbounded function.
Hence, I think a different approach is needed.
Thank you.