Exercise 7.11

Question

Exercise 11: 
    Suppose that $\{f_n\}$, $\{g_n\}$ are defined on $E$, and 
    \begin{itemize}
        \item[(a)] $\sum f_n$ has uniformly bounded partial sums;
        \item[(b)] $g_n\rightarrow 0$ uniformly on $E$;
        \item[(c)] $g_1(x)\ge g_2(x)\ge g_3(x)\ge\cdots$ for every $x\in E$.
    \end{itemize}
    Prove that $\sum f_ng_n$ converges uniformly on $E$.

analambanomenos · Accepted Answer

\def\eps{\varepsilon}

Theorem 3.42 gives us pointwise convergence of the series. Uniform convergence follows by repeating the proof of that Theorem in this context, as follows.

Let $F_n(x)$ be the partial sums of $\sum f_n(x)$. Choose $M$ such that $\big|F_n(x)\big|<M$ for all $n$ and all $x\in E$. Given $\varepsilon>0$ there is an integer $N$ such that $g_N(x)\le
    \varepsilon/(2M)$ for all $x\in E$. For $N\le p\le q$ we have by Theorem 3.41 
    \begin{align*}
        \Bigg|\sum_{n=p}^qf_n(x)g(x)\Bigg| &= \Bigg|\sum_{n=p}^{q-1}F_n(x)\bigl(g_n(x)-g_{n+1}(x)\bigr)+F_q(x)g_q(x)-F_{p-1}(x)g_p(x)\Bigg| \
        &\le M\Bigg|\sum_{n=p}^{q-1}\bigl(g_n(x)-g_{n+1}(x)\bigr)+g_q(x)+g_{p}\Bigg| \
        &= 2Mg_p(x)\le 2Mg_N(x)\le\varepsilon.
    \end{align*}
    Uniform convergence follows from Theorem 7.8 (the Cauchy condition for uniform convergence).

Exercise 7.11

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Exercise 7.11

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