On $\displaystyle [a,b]$, the function sequence $\displaystyle \{f_n\},\{g_n\}$ converge uniformly to $\displaystyle f,g$ respectively. Suppose there exists positive sequence $\displaystyle M_n$ such that $\displaystyle f_n(x)\leq M_n, g_n(x)\leq M_n,\ \forall\ x\in [a,b]$. Prove that $\displaystyle f_ng_n$ converge unformly to $\displaystyle fg$ on $\displaystyle [a,b]$

PS: If $\displaystyle M_n=M$, I know how to prove. But this?....Would you help me?