Dot Product with Self is Zero iff Zero Vector/Proof 2

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

Then:


 * $\mathbf u \cdot \mathbf u = 0 \iff \mathbf u = \mathbf 0$

Proof
Let $u\cdot u = 0$.

Then:

The only way for this to happen is if:
 * $\left\Vert{ \mathbf u }\right\Vert = 0$

which implies:
 * $\mathbf u = \mathbf 0$

Now suppose $\mathbf u = \mathbf 0$.

Then: