I'm attempting to understand a proof of a weak version of the central limit theorem. I got through the proof, but it relied on a lemma that was left without a proof nor without any motivation. It is of course the lemma in the picture below. How does one prove this? Does anyone perhaps know where I may find a proof of it that is suited for someone who has had calculus and real analysis on an undergraduate level? It does not have to be super rigorous, as long as it's sufficient on some basic level.