For the paper I am working on, I am trying to prove that the Cauchy sequence of semicontinuous functions converges to semicontinuous function (upper to upper, lower to lower). In other words, I am trying to prove that the space of bounded semicontinuous functions is a complete metric space (with the usual sup norm).

I think I can prove it but I am sure it must be somewhere in the literature. Can someone please suggest a reference where the proof is given as I find it pointless to include it in the paper just for the sake of having another proof there.

Thank you