Prove the theorem

Question

Please help me proof the theorem: If $A$ is compact operators, then

$a)$ (The Fredholm Alternative) $\sigma(A)=\{0\}\cup\Pi_{0}(A)$

$b)$ $\Pi_{0}(A)$ is either finite consist of a sequence converging to 0.

$c)$ The eigenspace $\{x:Ax=\lambda x\}$ is a finite-dimensional if $\lambda\neq0$.