# Uniform convergence of sequence of functions

## Homework Statement

Let $f_{n}(x)=\frac{x}{1+x^n}$ for $x \in [0,∞)$ and $n \in N$. Find the pointwise limit f of this sequence on the given interval and show that $(f_{n})$ does not uniformly converge to f on the given interval.

## The Attempt at a Solution

I found that the pointwise limit f is :

$0$ if $x=0$
$x$ if $0<x<1$
$1/2$ if $x=1$
$0$ if $x>1$

But I'm stuck on proving that it's not uniformly convergent. I know that for any natural number N, I need to find some x such that $|f_{n}(x) - f(x)| ≥ ε$, but I'm not sure how to go about this.

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pasmith
Homework Helper

## Homework Statement

Let $f_{n}(x)=\frac{x}{1+x^n}$ for $x \in [0,∞)$ and $n \in N$. Find the pointwise limit f of this sequence on the given interval and show that $(f_{n})$ does not uniformly converge to f on the given interval.

## The Attempt at a Solution

I found that the pointwise limit f is :

$0$ if $x=0$
$x$ if $0<x<1$
$1/2$ if $x=1$
$0$ if $x>1$

But I'm stuck on proving that it's not uniformly convergent. I know that for any natural number N, I need to find some x such that $|f_{n}(x) - f(x)| ≥ ε$, but I'm not sure how to go about this.
Look at $|f_n(x) - f(x)|$ for $x$ close to, but not equal to, 1.

$$\frac{x}{1+x^n}-x = \frac{x^{n+1} }{1+x^n} \geq \frac{x^{N+1} }{1+x^N} \geq x$$

Now choose x = e +1?