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

Theorem
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.

Let $T : \HH \to \HH$ be a bounded self-adjoint operator.

Let $\map \sigma T$ be the spectrum of $T$.

Then:
 * $\map \sigma T \subseteq \R$

Proof
This follows from:
 * Spectrum of Self-Adjoint Densely-Defined Linear Operator is Real and Closed
 * Spectrum of Bounded Linear Operator equal to Spectrum as Densely-Defined Linear Operator

Proof 2
For all $\phi \in \HH$ and $\lambda := a + i b \in \C$:

From this follows that, if $b \ne 0$, then $A - \lambda I$ is invertible, i.e. $\lambda \not \in \map \sigma T$.