Eigenvalues of Hermitian Operator are Real

Theorem
Let $H$ be a Hilbert space.

Let $A \in B \left({H}\right)$ be a self-adjoint operator.

Then all eigenvalues of $A$ are real.

Proof
Let $\lambda$ be an eigenvalue of $A$.

Let $v \in H$ be an eigenvector for $\lambda$; i.e. $Av = \lambda v$.

Now compute:

Now $v$, being an eigenvector, is non-zero.

By property $(5)$ of an inner product, this implies $\left\langle{v, v}\right\rangle \ne 0$.

Thence, dividing out $\left\langle{v, v}\right\rangle$, obtain $\lambda = \bar \lambda$.

From Complex Number Equals Conjugate iff Wholly Real, $\lambda \in \R$.