1617-257/TUT-R-2: Difference between revisions

From Drorbn
Jump to navigationJump to search
(Created page with "We discussed the following on 9/22/16: (1) What are the dimensions of <math>\mathbb{R}^\infty</math> and <math>\mathbb{R}^\omega</math>? (2) Let <math>S</math> be a subset o...")
 
No edit summary
Line 11: Line 11:
A student gave an for problem (2) which works fine if <math>S</math> is a closed set (it depended on the fact that <math>S' \subset S</math>).
A student gave an for problem (2) which works fine if <math>S</math> is a closed set (it depended on the fact that <math>S' \subset S</math>).


Let <math>\epsilon > 0</math> be given and let <math>x'' \in S''</math> be given.
Here's what we would have done if we had extra time to discuss:


Let <math>\epsilon > 0</math> be given.
Then there is some <math>x' \in S'</math> such that <math>\|x'' - x'\| < \epsilon/2.</math>


Let <math>\{x_k\}_{k \in \mathbb{N}} \subset S'</math> be a sequence which converges to <math>x</math>.
There is also a point <math>x \in S</math> such that <math>\|x' - x \| < \epsilon/2.</math>


Then there is some <math>n</math> for which <math>\| x_n - x\| < \epsilon/2</math>.
So <math>\|x - x''\| \leq \|x - x'\| + \|x' - x''\| < \epsilon.</math>


⚫
That is, <math>x \in S'.</math>
Since <math>x_n \in S'</math>, there is some <math>s \in S</math> such that <math>\|x_n - s\| < \epsilon/2</math>.

So <math>\|s - x\| \leq \|s - x_n \| + \|x_n - x\| < \epsilon.</math>

⚫
That is, <math>x \in S'</math>.

Revision as of 11:57, 30 September 2016

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]).

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]

Retrieved from "https://drorbn.net/index.php?title=1617-257/TUT-R-2&oldid=15383"