# 1617-257/TUT-R-2

We discussed the following on 9/22/16:

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

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

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

[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 ${\displaystyle \epsilon >0}$ be given and let ${\displaystyle x''\in S''}$ be given.

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

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

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

That is, ${\displaystyle x\in S'.}$