So, this is really late, the semester took its toll on me, and again nothing works.
The first problem is an application of the Eberlain-Smulyan theorem; let be a countable dense subset of and note that for each there exists a subsequence such that is weakly convergent. We choose the subsequences so that and form a diagonal sequence from these and call it .Let and and let . Choose large enough so that
and large enough so that
Then it follows that
The trick here was to use the definition to write it out rigorously, hence bypassing the need for dealing with limits.
For the second problem:
We first take a subsequence which we do not re-label so that where . We then use Mazur’s lemma to write down a convex combination of the , call them that converge strongly to for every . This is important that we create these sequences truncating off the first elements, since then this will allow us to pass the limit in the inequalities we will later obtain. More precisely, there exists with such that converges strongly to and so for every ,
Then we can first pass the limit and then pass the limit and obtain the result.