Iam not sure of the proof.

You can directly use DCT to prove this. given that function is bounded and has a pointwise convergence, first prove that f belongs to L' and lfl belongs to L'. that proof is obvious,as the functions are positive Then proceed with your method,use Fatou's lemma here both sup and inf and given the condition of function g, take limits k to infinity on both sides, and use the condition convergence of E(g), and you get the convergence of E(f).

I hope it helps