Section 21: Problem 6 Solution

Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James R. Munkres

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

For every $x<1$ , $f_{n}(x)\rightarrow f(x)=0$ , and $f_{n}(1)\rightarrow f(1)=1$ , but for every $\epsilon>0$ and points $x<1$ sufficiently close to $1$ , $f_{n}(x)>\epsilon$ (something like $(1-\delta)^{n}\ge1-n\delta$ , which can be proved by induction, might help). Another way to argue that $(f_{n}(x))$ does not converge uniformly is by using Theorem 21.6, as $f$ is not continuous.

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Section 21: Problem 6 Solution

Section 21: Problem 6 Solution