0
$\begingroup$

Define $f_n:[0,1]->\mathbb{R}$ by $f_n(x)=x^n$. Show that the sequence $(f_n(x))$ converges for each $x \in [0,1]$ but the sequence $(f_n)$ does not converge uniformly.

  • 0
    Hint for a slick proof: If a sequence of continuous functions converges uniformly what can you say about the limit function? What is the limit function here? Hint for a direct proof: What is the definition of uniform convergence? Draw some pictures and see where things go wrong.2012-02-13

2 Answers 2

4

HINTS:

  1. What is $\lim\limits_{n\to\infty}f_n(x)$ if $0\le x<1$?

  2. What is $\lim\limits_{n\to\infty}f_n(1)$?

  3. Even if $n$ is large, there must be values of $x<1$ such that $f_n(x)$ is close to $1$; why?

  4. For $x$ as in (3), $f_n(x)$ is not close to $0$; why does this matter?

2

Hint:

enter image description here

Note:

  • For $N$ a fixed, positive integer: $\lim\limits_{x\rightarrow 1^-} f_N(x) =1$.
  • But, for any $x\ne1$, we have $\lim\limits_{n\rightarrow \infty} f_n(x) =0$.