let $f_n(x) = \frac{x^n}{1 + x^n}$. determine whether $f_n \to f$ uniformly on $[0, 1]$ and whether $f_n \to f $ uniformly on $[0, \infty)$

Question

this sequence was given as a practice problem and I'm really having trouble. Heres the question:

My Work: I determined that the pointwise limit $f(x) = $lim$f_n(x)$ has to be $1$ but I am not too sure what to do after this. Any help is appreciated

Try the technique in my answer. – Mhenni Benghorbal Apr 08 '14 at 04:08 — Mhenni Benghorbal, Apr 08 '14 at 04:08

Pedro · Answer 1 · 2014-04-08T04:10:15.320

2

Is the pointwise limit a continuous function? Each function is continuous, so if convergence were uniform, the limit would be too.

Ans No, it is not. It is $0$ in $[0,1)$, $1/2$ at $1$ and $1$ over $(1,\infty]$.

edited Apr 08 '14 at 04:10

answered Apr 08 '14 at 03:46

Pedro

122,002

the question doesnt specify. All that it says is that it is a sequence – user3182418 Apr 08 '14 at 03:48
@user3182418 I am asking you. Is the pointwise limit a continuous function? – Pedro Apr 08 '14 at 03:48
This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. – user127096 Apr 08 '14 at 04:07
@cheapeffectivedietpills It does. – Pedro Apr 08 '14 at 04:10

score 0 · Answer 2 · answered Apr 08 '14 at 03:49

0

you can use the following characterization : $f_n $ converges uniformly to $f$ on $S$ iff

$$ \lim_{n \to \infty} \sup_{x \in S} \{ |f_n(x) - f(x) | \} = 0$$

answered Apr 08 '14 at 03:49

let $f_n(x) = \frac{x^n}{1 + x^n}$. determine whether $f_n \to f$ uniformly on $[0, 1]$ and whether $f_n \to f $ uniformly on $[0, \infty)$

2 Answers2