Dot Product of Vector with Itself/Proof 1

Theorem
Let $\mathbf u$ be a vector in the real vector space $\R^n$.

Then $\mathbf u \cdot \mathbf u = \left\|{ \mathbf u }\right\|^2$ where $\left\|{ \mathbf u }\right\|$ is the length of $\mathbf u$.

Proof
Let $\mathbf u = \left({ u_1, u_2, \ldots, u_n }\right)$. Then