In an answer to this question, a "long" proof is given. But the question can be equivalently stated, due to open mapping theorem, in the following manner.
Let $X$ be a Banach space and $M\subset X$ a closed subspace. Then, for every compact set $K$ in the quotient space $X/M$, there exists a compact set $L\subset X$ such that $\pi(L)=K$, where $\pi: X\to X/M$ is the quotient map.
Is there any short proof for this? Since the quotient map has many nice properties.