Spectrum of Self-Adjoint Bounded Linear Operator is Real and Closed/Proof 2

Proof
For all $\phi \in \HH$ and $\lambda := a + i b \in \C$:

Let $b \ne 0$.

In view of $(1)$, both $A - \lambda I$ and $A - \overline \lambda I$ are injective.

Moreover:

By Linear Subspace Dense iff Zero Orthocomplement, $\Img {A - \lambda I}$ is dense in $\HH$.

Now applying $(1)$ again, we can conclude that $A - \lambda I$ is invertible.

That is, $\lambda \not \in \map \sigma T$.