1617-257/TUT-R-2 - Revision history

Jeffim at 16:04, 30 September 2016

2016-09-30T16:04:12Z

← Older revision		Revision as of 12:04, 30 September 2016
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>).

			[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.]
	[The following proof was changed. It's essentially the same as before, but a different definition for limit point has been used.]

	Let <math>\epsilon > 0</math> be given and let <math>x'' \in S''</math> be given.		Let <math>\epsilon > 0</math> be given and let <math>x'' \in S''</math> be given.

Jeffim at 15:59, 30 September 2016

2016-09-30T15:59:52Z

← Older revision		Revision as of 11:59, 30 September 2016
Line 10:		Line 10:

	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>).

			[The following proof was changed. It's essentially the same as before, but a different definition for limit point has been used.]

	Let <math>\epsilon > 0</math> be given and let <math>x'' \in S''</math> be given.		Let <math>\epsilon > 0</math> be given and let <math>x'' \in S''</math> be given.

Jeffim at 15:57, 30 September 2016

2016-09-30T15:57:26Z

← Older revision		Revision as of 11:57, 30 September 2016
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>.

Jeffim: Created page with "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 o..."

2016-09-23T15:17:44Z

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..."

New page

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 of <math>\mathbb{R}^n</math>. Show that the set of limit points of <math>S</math>, <math>S'</math>, is closed.

----

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>).

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

Let <math>\epsilon > 0</math> be given.

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

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

1617-257/TUT-R-2 - Revision history

Jeffim at 16:04, 30 September 2016

Jeffim at 15:59, 30 September 2016

Jeffim at 15:57, 30 September 2016

Jeffim: Created page with "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 o..."

Jeffim: Created page with "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 o..."