This implication is true, but not obvious. Can you justify it? (or maybe it is from your lecture notes)

Nothing new is needed. Just prove that the sequence satisfies the assumptions of question 1) (with 0 as the limit).2) But now I am stuck with question 2. The assumption is replaced by (Xn)^2 ≤ W, and we need to prove more: Xn->X in mean square. I tried to modify the proof in question 1, but it doesn't seem to work here. How can we modify the proof? Or do we need something new?

Hint: using the property about subsequences, you have as well.