The following problem is from the spring $2020$ analysis qualifying exam given at UCLA:
Define $T_n\in L^{\infty}(\mathbb{R})^*$ as $$T_n(f)=\int_0^\infty \frac{x^n}{n!}e^{-x}f(x)dx$$
- Prove that no subsequence of this sequence converges weak-∗.
- Explain why this does not contradict the Banach–Alaoglu Theorem.
The second point is not too hard: $||T_n||=1$ and it does not contradict Banach-Alaoglu because Banach-Alaoglu gives compactness of the unit ball, not sequential compactness, in general. We have sequential compactness if the banach space is separable, and $L^\infty(\mathbb{R})$ is not so there's no contradiction.
However, I am stumped by the first point: my first idea was to try and find, given a sequence $T_{n_k}$, a function $f$ such that $T_{n_k}(f)$ does not converge (perhaps by some diagonalization process) but I was not able to construct such a function.
Any hint is appreciated