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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.

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'.