measurable functions and convergence

Prove that the set of points at which a sequence of measurable real functions converges is a measurable set.

I have an approach, but it doesn't seem clear to me:
Denote S(n) the set of x for which the inequality |f_k(x)-f_m(x)|<1/n holds for k,m > some natural number.
Then if g_{km}(x)=f_k(x)-f_m(x)-measurable then the set of x for which |g_{km}(x)|<1/n is measurable and so does the union of such sets when k,m goes from some natural number to \infty. So S(n) is measurable.

Then the set in the problem condition of those x at which sequence {f_n} converges is the intersection of S(n) when n goes from 1 to \infty, thus it is measurable(as a countable intersection of measurable sets ).

Thank you for reading this. Is it ok? Or maybe there is more clear solution for this (i think very easy for most of you )problem.
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