I would be grateful if you can give me some hints for the following homework problem.
Let $C$ be a subset of $l^\infty$ (with uniform norm) such that $C = \left\{(x_n) \mid |x_n|\leq \frac1n \,\forall n\geq 1\right\}$ Is $C$ a compact set or not?
Honestly, I am stuck. I tried to use sequentially compactness and attempted to construct different sequences which may serve as counter examples but all of them failed to do so.
I also thought that if $y_n$ is a constant sequence equal to $0$ and if we take the open ball $B(y_n,2)$ with radius $2$, does that count as a finite open cover of $C$?
Thanks in advance.