1617-257/TUT-R-2

We discussed the following on 9/22/16:

(1) What are the dimensions of $\mathbb {R} ^{\infty }$ and $\mathbb {R} ^{\omega }$?

(2) Let $S$ be a subset of $\mathbb {R} ^{n}$. Show that the set of limit points of $S$, $S'$, is closed.

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A student gave an for problem (2) which works fine if $S$ is a closed set (it depended on the fact that $S'\subset S$).

[A student pointed out that I used a definition for limit point which was different from (but also equivalent to) that given in the text. We've replaced any usage of the definition I originally used with the text's definition. We also discussed why the two definitions are equivalent in the Thursday tutorial that took place on 9/29/16.]

Let $\epsilon >0$ be given and let $x''\in S''$ be given.

Then there is some $x'\in S'$ such that $\|x''-x'\|<\epsilon /2.$

There is also a point $x\in S$ such that $\|x'-x\|<\epsilon /2.$

So $\|x-x''\|\leq \|x-x'\|+\|x'-x''\|<\epsilon .$

That is, $x\in S'.$