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 $S'\subset S$).

Here's what we would have done if we had extra time to discuss:

Let $\epsilon >0$ be given.

Let $\{x_{k}\}_{k\in \mathbb {N} }\subset S'$ be a sequence which converges to $x$.

Then there is some $n$ for which $\|x_{n}-x\|<\epsilon /2$.

Since $x_{n}\in S'$, there is some $s\in S$ such that $\|x_{n}-s\|<\epsilon /2$.

So $\|s-x\|\leq \|s-x_{n}\|+\|x_{n}-x\|<\epsilon .$

That is, $x\in S'$.