Talk:Proofs of convergence of random variables

Possible error in lemma in proof that convergence in probability implies convergence in distribution

I feel like there's an error in the proof of said lemma where it says that the expressions on the fourth and the fifth line are equal. Seeing as |X - Y| > ε implies ( X - Y > ε or X - Y < -ε ), but not the other way around. I didn't feel confident enough in my knowledge of probability theory to correct it though.