$(a)$ Using the fact that squares are never negative show for any real numbers $p$ and $q$:$p^2+q^{2}2pq.$

$(b)$ Suppose $f(x)$ and $g(x)$ are continuous, positive functions defined over $[a,).$ Prove the

following statement:

$If{\displaystyle \underset{a}{\overset{}{}}}(f(x){)}^{2}dx$ and ${\displaystyle \underset{a}{\overset{}{}}}(g(x){)}^{2}dx$ are both convergent

then ${\displaystyle \underset{a}{\overset{}{}}}f(x)g(x)dxis$ also convergent.