Uniform convergence of sequence of functions

[phosgene]

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.

Homework Equations

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.

 

[pasmith]

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.

Homework Equations

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.

 

[dirk_mec1]

\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?