Prove This Set is Measurable

Hey all,

Problem: Let $\displaystyle \{f_n\}$ be a sequence of measurable functions. Prove that the set of points at which $\displaystyle \{f_n\}$ converges is a measurable set.

My Attempted Solution: We know that $\displaystyle \limsup f_n$ and $\displaystyle \liminf f_n$ are measurable functions; hence $\displaystyle g = \limsup f_n - \liminf f_n$ is measurable. The set of values at which $\displaystyle \{f_n\}$ converges is equal to $\displaystyle \{x: g(x) = 0\}$ which is measurable because $\displaystyle g$ is measurable. Q.E.D.

Does this work? I tried to check my answer online, but the methods I saw differ pretty significantly from this, which makes me question whether or not this works.