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 {x_k}_{k\in\mathbb{N}} be a countable dense subset of X and note that for each k\in \mathbb{N} there exists a subsequence \Lambda_k\subset \mathbb{N} such that (F_n(x_k))_{n\in\Lambda_k} is weakly convergent. We choose the subsequences so that \Lambda_1\subset\Lambda_2\subset\dots and form a diagonal sequence from these and call it \Lambda.Let l\in Y^* and x\in X and let \eps>0. Choose k\in \mathbb{N} large enough so that


and n,m\in\Lambda large enough so that

    \[|l(F_n(x_k))-l(F_m(x_k))|< \eps.\]

Then it follows that

    \[\abs{l(F_n(x))-l(F_m(x))}\leq \eps.\]

The trick here was to use the \eps 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 F(x_k)\to\alpha_0 where \alpha_0:=\liminf_{k\to\infty}F(x_k)\leq - \infty. We then use Mazur’s lemma to write down a convex combination of the (x_k)_{k\geq k_0}, call them (y_k)_{k\geq k_0} that converge strongly to x for every k_0\in\mathbb{N}. This is important that we create these sequences truncating off the first k_0 elements, since then this will allow us to pass the limit in the inequalities we will later obtain. More precisely, there exists 0 \leq \alpha_{k,l} \leq 1 with \sum_{l=k_0} \alpha_{k,l} = 1 such that y_k :=\sum_{l\in\mathbb{N}}\alpha_{k,l}x_l converges strongly to x and so for every k\geq k_0,

    \[F(y_k)\leq \sum_{l=k_0} \alpha_{k,l}x_l \leq \sup_{l\geq k_0} F(x_l).\]

Then we can first pass the limit k\to \infty and then pass the limit k_0\to\infty and obtain the result.

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