If $A \subseteq l_\infty$, and $A=\{l\in l_\infty: |l_n| \le b_n \}$, where $b_n$ is a sequence of real, non-negative numbers, then if $A$ is compact subset of $X$ it must mean that $\lim (b_n) = 0$.
I tried doing this by contradicition, if $A$ is compact, it means that it is closed subset in $X$, which implies it is complete, but if we assume $\lim(b_n) \neq0$ I should maybe be able to show $\exists$ a Cauchy sequence for which this sequence converges outside of $A$. However, I can't think of any counterexample. Am I doing this wrong?