Prove the compactness of an operator in a Hilbert space

Question

I'm trying to prove the following statement:

Let $\{M_k, k \in \mathbb{N} \}$ be a countable collection of finite dimensional subspaces of a Hilbert space $H$. For every $k$ let $P_k$ be the orthogonal projection operator on $M_k$. Prove that the operator $$P = \sum_{k \in \mathbb{N}} 2^{-k} P_k$$ is compact.

I know that one can easily prove that each $P_k$ is compact, but I can't figure out how to procede from here. Any hint would be really appreciated.

Answer 1

Hint: $P=lim_n\sum_{k=1}^n2^{-k}P_k$ and $P_n=\sum_{k=1}^n2^{-k}P_k$ is compact, to see this remark that $\|P_k\|=1$ and $\|\sum_{k\geq n}2^{-k}P_k\|\leq\sum_{k\geq n}2^{-k}\|P_k\|=\sum_{k\geq n}2^{-k}$.