Uniform convergence of sum of sequence

Question

Please help me prove the following:

Suppose that $(f_n)$ and $(g_n)$ are sequences of real valued functions on a set E such that

1. There is a real A such that $|f_0(x)+...+f_n(x)|$ < A $\forall n$, $\forall x$
2. $g_n \to 0$ uniformly as $n \to \infty$
3. $g_0(x) \geq g_1(x) \geq ... $ for every x in E

Prove that $f_0 g_0 + f_1 g_1 + ... + f_n g_n$ converges uniformly on E as $n \to \infty$

My solution attempt:

$S_n = g_0 f_0+ g_1 f_1 + ... + g_n f_n$. An Abel transformation gives the following.

$S_n = g_n F_n - \sum_{k=0}^{n-1} F_{k} (g_{k+1}-g_{k})$

where $F_n = f_0 + ... + f_n$

We see that $F_n$ is bounded according to condition (1) and if $n \to \infty $ then $g_n \to 0$ according to (2). So $g_n F_n$ is bounded.

Moreover

$\sum_{k=0}^{n-1} F_{k} (g_{k+1}-g_{k}) \leq (\sum_{k=0}^{n-1} F_{k}) \times (\sum_{k=0}^{n-1}(g_{k+1}-g_{k})) = (\sum_{k=0}^{n-1} F_{k}) \times (g_n - g_0)$

Where $\sum_{k=0}^{n-1} F_{k}$ and $g_n - g_0$ are bounded according to (1) and (3) respectively. Since all expressions are bounded, the Cauchy criterion gives that $S_n$ converges uniformly as $n \to \infty$.

Answer 1

Hint

The proof is based on two ideas

• the summation by parts
• the Cauchy criterion for the uniform convergence of the series.