The Abel's test for uniform convergence states: Assume$\displaystyle \sum f_n(x)$ converges uniformly on E. A uniformly bounded sequence of functions $\displaystyle \{g_n(x)\}$ satisfies $\displaystyle g_{n+1}(x)\leq g_n(x)$ for all n and x in E. Then $\displaystyle \sum f_n(x)g_n(x)$ converges uniformly.

My tentative proof is as follows:

Let $\displaystyle s_n(x)=\sum\limits_{k = 1}^n f_k(x)g_k(x), F_n(x)=\sum\limits_{k = 1}^n f_k(x), x\in E$. By the partial summation formula, we have $\displaystyle s_n(x)=F_n(x)g_1(x)+\sum\limits_{k = 1}^n (F_n(x)-F_k(x))(g_{k+1}(x)-g_k(x))$. So if n>m we can get $\displaystyle s_n(x)-s_m(x)=(F_n(x)-F_m(x))g_1(x)+\sum\limits_{k = m+1}^n (F_n(x)-F_k(x))(g_{k+1}(x)-g_k(x))$. If the uniform bound for $\displaystyle \{g_n(x)\}$ is M, the absolute value of the above difference satisfies $\displaystyle |s_n(x)-s_m(x)|\leq M|F_n(x)-F_m(x)|+2M\sum\limits_{k = m+1}^n |F_n(x)-F_k(x)|$. From the fact that $\displaystyle \sum f_n(x)$ converges uniformly, for any given $\displaystyle \epsilon>0$, there is an N such that for all m,n>N we have $\displaystyle |F_n(x)-F_m(x)|<\epsilon$. So the right side of the above inequality $\displaystyle \leq [1+2(n-m)]M\epsilon$. But the variable (n-m) makes the Cauchy condition inapplicable. Can you tell me how to proceed, or is there any other better proof? Thanks!