This is problem 12 in Reed & Simon's book on functional analysis.
Let $\{\delta_n\}$ be the sequence in $\ell_\infty^*$ such that $$\delta_n\big(\{c_k\}_{k=1}^\infty\big) = c_n, \quad \forall \{c_k\}_{k=1}^\infty \in \ell_\infty.$$
Prove that $\{\delta_n\}$ has no weak-* convergent subsequence but has a weak-* convergent subnet.
I have been having a hard time getting started with this problem. There is already an answer posted on here for the subnet part of the problem (Showing that a sequence in $\ell^\infty(\mathbb{N})^*$ has a weak* convergent subnet), but it uses filters which I am familiar with and which Reed and Simon avoid.
Does anyone have any tips on how to get started?