1617-257/TUT-R-2

We discussed the following on 9/22/16:

(1) What are the dimensions of [math]\displaystyle{ \mathbb{R}^\infty }[/math] and [math]\displaystyle{ \mathbb{R}^\omega }[/math]?

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

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

[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 [math]\displaystyle{ \epsilon \gt 0 }[/math] be given and let [math]\displaystyle{ x'' \in S'' }[/math] be given.

Then there is some [math]\displaystyle{ x' \in S' }[/math] such that [math]\displaystyle{ \|x'' - x'\| \lt \epsilon/2. }[/math]

There is also a point [math]\displaystyle{ x \in S }[/math] such that [math]\displaystyle{ \|x' - x \| \lt \epsilon/2. }[/math]

So [math]\displaystyle{ \|x - x''\| \leq \|x - x'\| + \|x' - x''\| \lt \epsilon. }[/math]

That is, [math]\displaystyle{ x \in S'. }[/math]