Almost Sure Convergence Using Borel-Cantelli 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.

Assume that $f_n$ is Lebesgue almost everywhere finite, for every $n$ (otherwise the result fails). 
For every $n$, the interval $[-n,n]$ has finite Lebesgue measure hence there exists $c_n\gt0$ such that the Lebesgue measure of the Borel set 
$$
A_n=\{x\in[-n,n]\,\mid\,c_n\cdot|f_n(x)|\gt1/n^2\}
$$ 
is at most $1/n^2$.
Then,
Borel-Cantelli lemma shows that $\limsup A_n$ has Lebesgue measure zero, hence, for Lebesgue almost every $x$ in $\mathbb R$, $x$ is not in $A_n$ for every $n$ large enough. Since $|x|\leqslant n$ for every $n$ large enough, this means that $c_n\cdot|f_n(x)|\leqslant1/n^2$ for every $n$ large enough. 
In particular, the series $\sum\limits_nc_nf_n(x)$ converges (absolutely). QED.
Edit: Assume that there exists $k$ such that $f_k$ is not Lebesgue almost everywhere finite. This means that $A=\{x\in\mathbb R\,\mid\,|f_k(x)|=+\infty\}$ has positive Lebesgue measure. Now, for every positive valued sequence $(c_n)$, the series $\sum\limits_nc_nf_n$ diverges on $A$. Hence there exists no positive valued sequence $(c_n)$ such that the series $\sum\limits_nc_nf_n$ converges Lebesgue almost everywhere.
