Degenerate Linear Operator Plus Identity is Fredholm Operator

Theorem
Let $U$ be a vector space.

Let $T : U \to U$ be a degenerate linear operator.

Let $I_U : U \to U$ be the identity operator.

Then:
 * $T + I_U$

has finite index.

Proof
We need to show:
 * $\ds \map \dim {\map \ker {T + I_U} } < + \infty$

and:
 * $\ds \map \dim { U / {\Img {T + I_U} } } < + \infty$