Question about uniform convergence

**wontonsoup** · December 14th 2010, 04:18 PM

Hi, so Im studying for finals and am stuck on a problem. I got referred to this forum from a friend, and I had a quick question about uniform convergence.
we have a function which is:

$f_n(x)\frac{nx}{nx+1}$ for all n greater than or equal to 1.

Is this uniformly convergent at [0,1]?
Then if we were to fix r>0, is it uniformly convergent at [r, infinity]?

Also, there was another part of the problem. the professor wrote something like
$f(x) = \lim_{x\to\infty}f(x^n)$
He's a messy writer, and i do believe that was what he wrote, but Im not too sure if that fits into the problem at all, or if it even does.

Any help is appreciated. Thanks!

**FernandoRevilla** · December 14th 2010, 10:41 PM

Originally Posted by wontonsoup

$f_n(x)\frac{nx}{nx+1}$ for all n greater than or equal to 1. Is this uniformly convergent at [0,1]?

Every $f_n$ is continuous on $[0,1]$ , however:

$f(x)=\displaystyle\lim_{n \to \infty}f_n(x)=\begin{Bmatrix}0&\textrm{if}&x=0\\1& \textrm{if}&x\in(0,1]\end{matrix}$

is not continuous on $[0,1]$ , so by a well known theorem, the convergence is not uniform.

Then if we were to fix r>0, is it uniformly convergent at [r, infinity]?

Yes, now

$f(x)=\displaystyle\lim_{n \to \infty}f_n(x)=1$

Prove that for every $\epsilon>0$ there exists $n_0\in\mathbb{N}$ such that

$|f_n(x)-f(x)|<\epsilon$ for $n\geq n_0$ and for all $x\in[r,+\infty)$

Also, there was another part of the problem. the professor wrote something like
$f(x) = \lim_{x\to\infty}f(x^n)$
He's a messy writer, and i do believe that was what he wrote, but Im not too sure if that fits into the problem at all, or if it even does.

Possibly is:

$f(x)=\displaystyle\lim_{n \to \infty}f_n(x)$

Fernando Revilla

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