Defining Elements on the Real Line?

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RE-EDIT: I'm confused again, continue on reading Bonjourno, I'm trying to work at the lectures provided on youtube by nptelhrd but I've gotten my foot stuck in a hole in the real line only 15 minutres into it

(In my head I say "such that" whenever the symbol : pops up!).

We define a set;

A : {r ∈ Q: r²<2}

Does ∃ a largest element of A in Q?

1: We seek to find some n ∈ N : $$( r \ + \ \frac{1}{n} )$$ will satisfy the conditions specified by A.

2: $$(r \ + \ \frac{1}{n} )^2 \ < \ 2$$

3: $$r^2 \ + \ \frac{2r}{n} \ + \ \frac{1}{n^2} \ < \ 2$$

4: $$\frac{2r}{n} \ + \ \frac{1}{n^2} \ < \ 2 \ - \ r^2$$

The R.H.S. is strictly positive due to r²<2.

Okay, I understand up to here but then the lecturer starts to get confusing, he then says that It suffices only to find some n ∈ N :

$$\frac{2r}{n} \ + \ \frac{1}{n} \ < \ 2 \ - \ r^2$$

Notice the n and not n² on the bottom of the L.H.S. Fraction!

He says;

This is because;

[tex]\frac{1}{n^2} < \frac{1}{n} \ and \ this \ implies \ \frac{2r}{n} \ + \ \frac{1}{n^2} \ < \ \frac{2r}{n} \ + \ \frac{1}{n} [/itex]<br /> <br /> I have no idea where this came from!The video is on youtube <a href="http://www.youtube.com/watch?v=0lzOAW8yMTc&feature=PlayList&p=8F599D7DB30C539B&playnext_from=PL&index=0&playnext=1" target="_blank" class="link link--external" rel="nofollow ugc noopener">http://www.youtube.com/watch?v=0lzO...DB30C539B&playnext_from=PL&index=0&playnext=1</a> and I would say everything he is trying to do is described from 10:00 to 14:00.<br /> <br /> I would <b>extremely</b> appreciate it if someone could take 6 minutes to watch this and correct me as I have nobody else <img src="https://www.physicsforums.com/styles/physicsforums/xenforo/smilies/oldschool/redface.gif" class="smilie" loading="lazy" alt=":redface:" title="Red Face :redface:" data-shortname=":redface:" /> to explain it to me.<br /> <br /> What I think is going on is that he is trying to prove a least upper bound or something and that this will show that the real line can be continuously divided, or something.[/tex]

 

[HallsofIvy]

RE-EDIT: I'm confused again, continue on reading


Bonjourno, I'm trying to work at the lectures provided on youtube by nptelhrd but I've gotten my foot stuck in a hole in the real line only 15 minutres into it

(In my head I say "such that" whenever the symbol : pops up!).

We define a set;

A : {r ∈ Q: r²<2}

Does ∃ a largest element of A in Q?

1: We seek to find some n ∈ N : $$( r \ + \ \frac{1}{n} )$$ will satisfy the conditions specified by A.

2: $$(r \ + \ \frac{1}{n} )^2 \ < \ 2$$

3: $$r^2 \ + \ \frac{2r}{n} \ + \ \frac{1}{n^2} \ < \ 2$$

4: $$\frac{2r}{n} \ + \ \frac{1}{n^2} \ < \ 2 \ - \ r^2$$

The R.H.S. is strictly positive due to r²<2.

Okay, I understand up to here but then the lecturer starts to get confusing, he then says that It suffices only to find some n ∈ N :

$$\frac{2r}{n} \ + \ \frac{1}{n} \ < \ 2 \ - \ r^2$$

Notice the n and not n² on the bottom of the L.H.S. Fraction!

He says;

This is because;

[tex]\frac{1}{n^2} < \frac{1}{n} \ and \ this \ implies \ \frac{2r}{n} \ + \ \frac{1}{n^2} \ < \ \frac{2r}{n} \ + \ \frac{1}{n} [/itex]<br /> <br /> I have no idea where this came from![/tex]

[tex] For any integer n> 1, $1> \frac{1}{n}$ (divide both sides of n> 1 by the positive number n) so $\frac{1}{n}> \frac{1}{n^2}$ (divide both sides by n again). Now, adding any number a to both sides, $a+ \frac{1}{n}> a+ \frac{1}{n^2}$. In particular, if $a= \frac{2r}{n}$, $\frac{2r}{n}+ \frac{1}{n}> \frac{2r}{n}+ \frac{1}{n^2}$.<br /> <br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> The video is on youtube <a href="http://www.youtube.com/watch?v=0lzOAW8yMTc&feature=PlayList&p=8F599D7DB30C539B&playnext_from=PL&index=0&playnext=1" target="_blank" class="link link--external" rel="nofollow ugc noopener">http://www.youtube.com/watch?v=0lzO...DB30C539B&playnext_from=PL&index=0&playnext=1</a> and I would say everything he is trying to do is described from 10:00 to 14:00.<br /> <br /> I would <b>extremely</b> appreciate it if someone could take 6 minutes to watch this and correct me as I have nobody else <img src="https://www.physicsforums.com/styles/physicsforums/xenforo/smilies/oldschool/redface.gif" class="smilie" loading="lazy" alt=":redface:" title="Red Face :redface:" data-shortname=":redface:" /> to explain it to me.<br /> <br /> What I think is going on is that he is trying to prove a least upper bound or something and that this will show that the real line can be continuously divided, or something. </div> </div> </blockquote>[/tex]

 

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Thanks a lot HallsofIvy for clearing that up!