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.

---

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]).

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

Let [math]\displaystyle{ \epsilon \gt 0 }[/math] be given.

Let [math]\displaystyle{ \{x_k\}_{k \in \mathbb{N}} \subset S' }[/math] be a sequence which converges to [math]\displaystyle{ x }[/math].

Then there is some [math]\displaystyle{ n }[/math] for which [math]\displaystyle{ \| x_n - x\| \lt \epsilon/2 }[/math].

Since [math]\displaystyle{ x_n \in S' }[/math], there is some [math]\displaystyle{ s \in S }[/math] such that [math]\displaystyle{ \|x_n - s\| \lt \epsilon/2 }[/math].

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

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