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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;
\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.
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;
\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.
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