Let $X$ be a Banach space and $K$ a compact subset of $X$ and consider for a given $\eta>0$ the closed ball $C(0,\eta)$ centered at $0$ of radius $\eta$.
How can I show that $K+C(0,\eta)=\{x+y: x\in K~\mbox{and }y\in C(0,\eta)~\}$ is not (sequentially) compact (except when $X$ has finite dimension or $K$ is empty)?