Theorem 1   Let the real functions $f_1(x)$ , $f_2(x)$ , ... be defined in the interval $[a, b]$ . If they all satisfy the condition $$|f_n(x)| \leqq M_n \quad \forall\,x\in[a, b],$$ with $\sum_{n = 1}^{\infty}M_n$ a convergent series of constant terms, then the function series $$f_1(x)\!+\!f_2(x)\!+\!\cdots$$ converges uniformly on the interval $[a, b]$ .