Products of uniformly convergent uniformly bounded sequences

Let \((f_n)\) and \((g_n)\) converge uniformly on a set \(A\). Suppose both sequences are uniformly bounded: there is \(M\) such that \(|f_n(z)|,|g_n(z)|\le M\) for every \(z\in A\) and \(n\). Prove that \((f_ng_n)\) converges uniformly on \(A\).