A Self-Adjoint Operator on a Finite-Dimensional Space has a Unit Eigenvector

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space with zero vector $0_{V}$, and suppose that $V$ is finite-dimensional and $V\ne\{0_{V}\}$. Let $T$ be a linear operator on $V$ that is self-adjoint.

Then there are a unit vector $x_{0}\in V$ and a real number $\lambda$ such that $x_{0}$ is an eigenvector of $T$ with eigenvalue $\lambda$.