Is $K_n$ compact in $l^{\infty}(\mathbb{R})$ Consider $X=l^{\infty}(\mathbb{R})$ the space 
of all bounded sequence of real numbers endowed with the sup norm. 
I would like to know two thinks about this  Banach space:


*

*Are  the sets $K_n=[-n,n]^{\mathbb{N}}$  compacts?

*Has the  sets $K_n=[-n,n]^{\mathbb{N}}$ non empty interior?
Note: I'd like to know this thinks to overlap some technical
issues in a problem that I'm working.  

Note that $K_{n}$ is the closed ball centered at 0 and with radius $n$ in your space. So it has a non-empty interior. It is not compact: the sequences $s_{n}$ such that $s_{n}(m) = \delta_{n,m}$ are a sequence in your space that has no convergent subsequence, because $\lVert s_{n}-s_{m} \rVert = 1$
