Hi,

I have the following problem. Prove that if weakly AND , then strongly. (i.e. .

My proof so far.

Since weakly . I also know that . I think it is useful to use this hat notation because I know that . So I'll start with

something.

Details missing. Then,

.

Therefore, .

I just need help filling in the middle details. I was thinking of adding and subtracting something, but I just can't figure it out.

The next part of the question says find a counterexample to show that weakly does not imply .

I am bad with counterexamples. I know I need to come up with a sequence that converges weakly, which means I need a function such that this sequence converges, but I cannot think of one. Thank you for any ideas.