Let $H$ be a Hilbert space and let $T \in \mathcal{B}(H)$.

(a) Show that if $\|I-T\|<1$ then $T$ is invertible.

(b) Prove that if $T$ is invertible and if $S \in \mathcal{B}(H)$ satisfies $\|S-T\|<\left\|T^{-1}\right\|^{-1}$, then $S$ is invertible.

(c) Define what it means for $T$ to be compact. Prove that the set of compact operators on $H$ is a closed subset of $\mathcal{B}(H)$.

(d) Prove that $T$ is compact if and only if there is a sequence $\left(F_{n}\right)$ in $\mathcal{B}(H)$, where each operator $F_{n}$ has finite rank, such that $\left\|F_{n}-T\right\| \rightarrow 0$ as $n \rightarrow \infty$.

(e) Suppose that $T=A+K$, where $A$ is invertible and $K$ is compact. Prove that then, also, $T=B+F$, where $B$ is invertible and $F$ has finite rank.