Prove This Set is Measurable

Hey all,

Problem: Let be a sequence of measurable functions. Prove that the set of points at which converges is a measurable set.

My Attempted Solution: We know that and are measurable functions; hence is measurable. The set of values at which converges is equal to which is measurable because 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.