Real Analysis | The density of Q and other consequences of the Axiom of Completeness.

We present three results that follow from the completeness of the real numbers.

1. The Nested Interval Theorem
2. The Archimedean Principal
3. The density of the rational numbers in the real numbers.

Books I like:

Abstract Algebra:

Differential Forms:

Number Theory:

Calculus:

[tact]

Finally, a channel with some real mathematics.

[Bermatematika]

For the density of the rationals, my favorite argument is as follows: Let says you take your n such that 1/n <b-a as in @15:09. Now wlog 0<a<b and think of the segment (a, b) as a trench of length b-a. You are walking starting from 0 with step size 1/n. Since your step size is smaller than the trench, you can't jump over the trench, hence after m steps for some m you will fall into the trench, i.e. m/n should be in the interval (a, b). Hence a<m/n <b . We can formalize this by taking m to be the smallest m such that a <m/n. If on the contrary b <m/n, then m/n -1/n > b -(b-a) =a, so (m-1)/n >a contradicts the minimality of m.

[jaredvv]

This was explained extremely well. Keep them coming please

[jonaskoelker]

Theorem (used at 14:50): for all x in R there's a unique m in Z such that m <= x < m + 1.

Proof of existence:
Consider A = { a in Z | a <= x }.
Clearly x is an upper bound on A.
Also A is non-empty: there exists an n in N such that n > -x implying x > -n and -n is in Z.
Let m = sup(A). Clearly m <= x. If m + 1 <= x then m + 1 <= m since m is an upper bound, but then 1 <= 0 contradicting 1 > 0. So x < m + 1.

Proof of uniqueness:
Assume r in Z with r <= x < r + 1 and m defined as above.
Since r <= x we get r in A and thus r <= m since m is an upper bound on A.
But if r < m then r + 1 <= m (since they're integers) so r + 1 <= m <= x < r + 1, a contradiction.
So r = m.

[DarkMonolth]

You can't just release 900 videos a day on complex math topics!
Michael: Hold my beer

[catherineferris]

thank you for your analysis videos! finally, a professor who EXPLAINS their steps! please continue making!

[adarshkumar]

Your channel is best for real analysis.
Keep upgrading

[Ngocan-kznr]

Thank you so much, professor. I am so grateful to find your channel.
In fact, I don't believe my school teacher really understands what she is teaching. She only says what is written in the book, and when I ask a question, she cannot explain.
Really Analysis is still a challenging subject for me, but I love it.

[atomic_soup_juice]

I wish I had some of this content when I was talking my real analysis classes in school

[karthikthiagarajan]

Wonderful flow. You explain some subtle points concerning the theorems as well. Are you planning to teach all of real analysis and upload them?

[quantabot]

Playlist length so far is 12hours...Thank you for your time Professor!

[joshbolton]

Great video. Amazing example for students on how to use sup and inf effectively.

[alex_]

9:59 There is an opinion that *1/n < y* violates the quantification domain n ∈ N, because natural numbers (unlike the rational numbers) have no division operation, and that it would be better to write *1 < ny* instead.

[사기꾼진우야내가죽여]

Monotone convergence theorem is also an important consequence of the axiom of completeness from which we can derive a lot of essntial theorems in real analysis.

[CallMeIshmael]

I'm getting very impressed by how wide a variety of topics you're putting videos up about. Are you teaching classes on all these things, or is this just a fun thing you're doing in your free time?

[davidjflorezr]

very glad to find this, will thoroughly enjoy my winter break now

[GoutamDAS-lswb]

Thank you very much sir for posting these lectures. I have started learning Real Analysis from your website.

[Nilius]

Great courses in mathematics! I enjoy showing your videos and I look forward with great interest to further helpful topics.

[oida]

What about half open intervals: [a, b)={r in R: a<=r<b} and (a, b]={r in R: a<r<=b}?

[erikb]

You should put links to these videos in your competition videos. I'm interested in these videos but didn't know they existed until you mentioned them in your Q&A video.