I was reading Diestel book (Absolutely Summing Operators) and it says:
"(...) let $(f_n)$ be any weak null sequence in $\mathcal{C}(K)$. Then $(f_n)$ is bounded and converges pointwise to zero."
I was trying to check this assertion but I am not convinced with my proof. My idea is based in the weak operator compactness of closed bounded sets which can be used to prove that the convergence is uniform in the functionals. Then, taking the supremum, the pointwise convergence follows. Is that correct?