Linear Algebra-Cauchy-Schwarz Inequality Proof by Direct Definition, Universal Statement,Implication

These are my lecture for University and College level students.

THEOREM Properties of the Dot Product
If u, v, and w are vectors in Rn and c is a scalar, then the properties listed below
are true.
1. u ∙ v = v ∙ u
2. u ∙ (v + w) = u ∙ v + u ∙ w
3. c(u ∙ v) = (cu) ∙ v = u ∙ (cv)
4. v ∙ v = !v!2
5. v ∙ v ≥ 0, and v ∙ v = 0 if and only if v = 0.