[SOLVED] Integrability of an infinite series of functions

**akolman** · November 30th 2008, 10:11 AM

I need to prove that if the sequence of functions $\{ f_n \}$
converges uniformly to $f$ on $[0,1]$ , and each $f_n$ is integrable, then the limit function $f$ is also integrable on [0,1].

Does this help at all??
Since $\{ f_n \}$ converges uniformly to $f$ , then
$\exists N$ such that $n>N \Rightarrow |f(x)-f_n(x)|<\varepsilon$

$\Rightarrow -\varepsilon<f(x)-f_n(x)<\varepsilon<br /> \Rightarrow f_n(x)-\varepsilon<f(x)<f_n(x)+\varepsilon$

**Mathstud28** · November 30th 2008, 02:37 PM

Originally Posted by akolman

I need to prove that if the sequence of functions $\{ f_n \}$
converges uniformly to $f$ on $[0,1]$ , and each $f_n$ is integrable, then the limit function $f$ is also integrable on [0,1].

Does this help at all??
Since $\{ f_n \}$ converges uniformly to $f$ , then
$\exists N$ such that $n>N \Rightarrow |f(x)-f_n(x)|<\varepsilon$

$\Rightarrow -\varepsilon<f(x)-f_n(x)<\varepsilon<br /> \Rightarrow f_n(x)-\varepsilon<f(x)<f_n(x)+\varepsilon$

First let us remark that $\lim_{N\to\infty}S_N=\sum_{n=1}^{\infty}a_n$ where $S_N$ represents the $N$ th partial sum. So now $\sum{a_n}\text{ converges}\leftrightarrow~S_N\text{ converges}$

1. So now we know that for a sequence $\left\{S_N\right\}$ to be convergent it must be montonic and bounded. Therefore $\exists{M}~\backepsilon\lim_{N\to\infty}S_N\leqsla nt{M}\stackrel{\text{in turn}}{\implies}\sum{a_n}\leqslant{M}$ . So for every point where $\sum{f_n(x)}$ converges we have by neccessity $\exists{M}\backepsilon~\sum{f_n(x)}\leqslant{M}$ . And if $\forall{x}\in[0,1]~\sum{f_n(x)}\stackrel{\text{uniformly}}\longright arrow{f}$ then $\forall{x}\in[0,1]\exists{M}\backepsilon~f\leqslant{M}$ . Thus on $[0,1]$ $f$ is bounded.

2. Before we start the next step we are given that $f_n(x)$ is Riemann integrable, thus bounded and continous.

3.Next we just use the theorem that if $\forall{x}\in[a,b]~~\sum{f_n(x)}\stackrel{\text{uniformly}}\longrigh tarrow{f}$ and $f_n(x)$ is continuous on $[a,b]$ then $f$ is continous on $[a,b]$ . So the fact that $\sum{f_n(x)}\stackrel{\text{uniformly}}\longrighta rrow{f}\wedge{f_n(x)}\in\mathcal{C}~~\forall{x}\in[0,1]$ tells us that $f$ is continuous on $[0,1]$

4. So we have shown that on $[0,1]$ $f$ is both bounded and continous. So it is Riemann integrable.

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