Trick is usually to describe limit in terms of unions, intersections of measurable sets. I mean this in order to show that the limit is measurable. for the rest, partition your domain in "enough" (compact) pieces for the vertical intervals [n, n+1). I think I remember Wikipedia had a proof.
assume your function is bounded and divide up the range into small intervals. for each interval [a,b] take the functionm tohave value a on the inverse image of that interval.....this gives you a simple function whichn lies within |b-a| of your function on that set.....of course the limit is only uniform a.e. since the fucntion is only essentially bounded and not bounded.