Inner Product Norm is Norm

Theorem
Let $V$ be an inner product space over a subfield $\Bbb F$ of $\C$.

Let $\norm {\, \cdot \,}$ denote the inner product norm on $V$.

Then $\norm {\, \cdot \,}$ is a norm on $V$.

Proof
Let us verify the norm axioms in turn.

Axiom $(\text N 2)$
Part $(2)$ of Properties of Semi-Inner Product.

Axiom $(\text N 3)$
Part $(3)$ of Properties of Semi-Inner Product.

Hence all the properties of a norm have been shown to hold.