Let $H$ be an infinite-dimensional, separable Hilbert space. Let $T$ be a compact linear operator on $H$, and let $\lambda$ be a non-zero, approximate eigenvalue of $T$. Prove that $\lambda$ is an eigenvalue, and that the corresponding eigenspace $E_{\lambda}(T)=\{x \in H: T x=\lambda x\}$ is finite-dimensional.

Let $S$ be a compact, self-adjoint operator on $H$. Prove that there is an orthonormal basis $\left(e_{n}\right)_{n \geqslant 0}$ of $H$, and a sequence $\left(\lambda_{n}\right)_{n \geqslant 0}$ in $\mathbb{C}$, such that (i) $S e_{n}=\lambda_{n} e_{n}(n \geqslant 0)$ and (ii) $\lambda_{n} \rightarrow 0$ as $n \rightarrow \infty$.

Now let $S$ be compact, self-adjoint and injective. Let $R$ be a bounded self-adjoint operator on $H$ such that $R S=S R$. Prove that $H$ has an orthonormal basis $\left(e_{n}\right)_{n \geqslant 1}$, where, for every $n, e_{n}$ is an eigenvector, both of $S$ and of $R$.

[You may assume, without proof, results about self-adjoint operators on finite-dimensional spaces.]