f & g Riemann integrable, show fg is integrable

**If f and g are both Riemann integrable on [a,b], prove that fg is also Riemann integrable.**

This is what I've got so far...

Let ||f||∞= sup{|f(x)|: x E [a,b]}

$\displaystyle M_i(f,P)$ = sup{f(x): $\displaystyle x_{i-1}$ ≤ x ≤ $\displaystyle x_i$}

$\displaystyle m_i(f,P)$ = inf{f(x): $\displaystyle x_{i-1}$ ≤ x ≤ $\displaystyle x_i$} where P is a partition of [a,b]

Let x,t E [$\displaystyle x_{i-1}, x_i$]

Then |f(x)g(x)-f(t)g(t)| ≤ |f(x)| |g(x)-g(t)| + |f(x)-f(t)| |g(t)| ≤ ||f||∞ [$\displaystyle M_i(g,P)-m_i(g,P)$] + [$\displaystyle M_i(f,P)-m_i(f,P)$] ||g||∞

Any help is appreciated!

[also under discussion in math links forum]