I'm reading Royden Chap 15. The last remark in section 1 reads:
For $X = l^{\infty}$, $B^*$, the closed unity ball of $X^*$, is not weak-* sequentially compact. Indeed the sequence $\{\psi_n\}\subset B^*$ defined for each $n$ by $$\psi_n(\{x_k\})=x_n \qquad \forall \{x_k\}\in l^{\infty}$$ fails to have a weak-* convergent subsequence.
I am having trouble seeing how $\{\psi_n\}$ has no weak-* convergent subsequence. I'm having trouble even stating what I want to show. I think what I want to show is: $$\exists (c>0 \text{ and }\hat{x}\in X^{**}) \, \forall(n,m\in\mathbb{N})\text{ s.t. }c<|\hat{x}(\psi_n)-\hat{x}(\psi_m)|$$