If $A$ is positive semi definite then prove that $$|u'Av|\leq (u'Au)^\frac 1 2 (v'Av)^\frac 1 2.$$
Work done:
$$|u'A^\frac 1 2 A^\frac 1 2 v|=\left|\langle A^\frac 1 2 u, A^\frac 1 2 v\rangle \right|\leq ||A^\frac 1 2 u|| \,||A^\frac 12 v|| =(u'A^\frac 1 2 u)( v'A^\frac 1 2 v)$$