Every Ordered Field Must Contain A Countably Infinite Subset (With Proof)
Channel: Amour Learning
Date: 2021-04-16

In this video, I prove that every ordered field must contain a countably infinite subset.

This problem can be found in Dr. Jirí Lebl's free open-access textbook: "Basic Analysis I: Introduction to Real Analysis, Volume I"
https://www.jirka.org/ra/

Any feedback on how these videos can be improved would be great. I will be constantly updating these videos to ensure that everything is accurate. Thank you all for being such a great community. All content is released under Creative Commons "No Rights Reserved", so please feel free to use this video however you want.

My name is Kody Amour, and I teach free undergraduate mathematics courses on YouTube. I'm currently developing three YouTube courses simultaneously: Discrete Math, Linear Algebra, Real Analysis, Topology, Number Theory and Complex Analysis. These courses correspond to playlists that I am constructing indefinitely.

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Credit to all of the authors of all of the resources below that greatly helped influence this video.

Free open-access online math textbooks (I only use textbooks from this resource, since everything is free for my students):
https://aimath.org/textbooks/approved-textbooks/

A free in-depth tutorial for SageMath: https://doc.sagemath.org/html/en/tutorial/

Learn LaTeX in 30 minutes: https://www.overleaf.com/learn/latex/Learn_LaTeX_in_30_minutes
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My favorite math-oriented YouTube channels (in no particular order):
Aleph 0: https://www.youtube.com/channel/UCzBjutX2PmitNF4avysL-vg
The Bright Side of Mathematics: https://www.youtube.com/channel/UCdwo4k1RQHTcq_-WS7Cazqg
Math Visualized: https://www.youtube.com/channel/UCnym_SvWIt_VfycYNt105aQ
Insights into Mathematics: https://www.youtube.com/channel/UCXl0Zbk8_rvjyLwAR-Xh9pQ
ScienceClic English: https://www.youtube.com/channel/UCWvq4kcdNI1r1jZKFw9TiUA
Stand-up Maths: https://www.youtube.com/channel/UCSju5G2aFaWMqn-_0YBtq5A
Zach Star: https://www.youtube.com/channel/UCpCSAcbqs-sjEVfk_hMfY9w
Numberphile: https://www.youtube.com/channel/UCoxcjq-8xIDTYp3uz647V5A
3Blue1Brown: https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw
MIT OpenCourseWare: https://www.youtube.com/channel/UCEBb1b_L6zDS3xTUrIALZOw
Khan Academy: https://www.youtube.com/channel/UC4a-Gbdw7vOaccHmFo40b9g
PBS Infinite Series: https://www.youtube.com/channel/UCs4aHmggTfFrpkPcWSaBN9g
Lex Fridman: https://www.youtube.com/user/lexfridman