An exercise about projections on Hilbert space

Question

Let $H$ be a Hilbert space with an orthonormal basis $\{v_{n}\}_{n=1}^{\infty}$. The C$^{*}$-algebra $K$, the set of all compact operators on $H$, is a non-unital C$^{*}$-algebra. Let $p_{n}$ be the projection (from $H$) onto s$pan\{v_{1}, v_{2}, ..., v_{n} \}$. Then $\{p_{n}\}$ is an increasing sequence of projections in $K$, then for any $x\in K$, can we verify: $$||p_{n}x-x||\rightarrow 0 ? ~~as~~n\rightarrow \infty$$

Answer 1

Yes. Prove it first for a rank-one operator. Then deduce that it works for finite-rank operators. Finally, use the fact that every compact is a norm-limit of finite-rank operators.