Identity Minus Compact Linear Operator on Banach Space has Index Zero

Theorem
Let $\Bbb F \in \set {\R, \C}$.

Let $\struct {X, \norm {\,\cdot\,}_X}$ be a Banach space over $\Bbb F$.

Let $I : X \to X$ be the identity mapping.

Let $C : X \to X$ be a compact linear operator.

Then $I - C$ is a Fredholm operator with the index:
 * $\map {\mathrm{ind} } {I - C} = 0$