Real Analysis | The monotone sequence theorem.

After introducing the notion of a monotone sequence we prove the classic result known as the monotone sequence theorem.

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[sinecurve]

This content really takes me back to my undergrad days. Cheers!

[chaoticoli]

Such a good explanation. It's a really straightforward proof, too, which is always nice. Some results in real analysis can be much more convoluted.

[GreenMeansGOF]

Today I learned a really cool corollary of this theorem. I am not sure if it is on the internet or in a book somewhere. If anyone knows about it, I would definitely appreciate knowing a good place to look.

Cor: If {a_n} is a monotonically decreasing sequence and {b_n} is a monotonically increasing sequence and a_n > b_n for each n, then each sequence converges.

Proof: a_1 ≥ a_n > b_n ≥ b_1 and we apply the Monotone Convergence Theorem to each sequence. QED

[jonaskoelker]

At 3:22 "Let's look at a monotonically increasing sequence that's bounded—let's say above, that's the important part"

Every increasing sequence is bounded below by its first element. Every decreasing sequence is bounded above by its first element. Bounded in both directions is thus equivalent to "bounded in the direction the sequence is moving in".

I guess if we let important = not yet proven, Michael's statement is technically correct (the best kind) ;-)

[doontz]

i love these Calc 1 proofs.
they are so elegant and I loved taking that course

[ethanbartiromo]

This was amazing! I don’t remember if I understood this in my reals class or not, but this definitely made me understand it as a refresher

[ndelo]

You are magic... I love the visual trick of cleaning the board by knocking it... I enjoy your work, Thank you, Sir.

[false]

These videos are of great help; thanks!

[drpkmath]

Nice way of showing the proof of it. Great work

[ianloree]

Does that mean the harmonic series isn’t bounded?

[mepoor]

Dont we have to prove for the case where n<or=N

[wojciechgrunwald]

Will you do some polish math olympiad tasks?

[mushroomsteve]

Does this hold in an arbitrary metric space? This proof relied on the completeness of R.

[tracyh]

2:06 In order to prove that "if a monotone sequence converges then it is bounded", you take the contrapositive statement, and then cite the contrapositive of "if a sequence converges then it is bounded". This is extremely unnecessary and silly! Just quote the previous video wholesale. You have already proved in the video that you quoted that any sequence that converges (regardless if it is monotone) is bounded.

[akihayakawa]

tysm ong it makes sooo much more sense than my professor taught me lol

[cloriscloris]

Yr explanation is really really gd but may I have these videos in a set{Real Analysis} s. t. I can find the video I nd quickly? Thank you

[jamesclerkmaxwell]

you forgot to show the case where an is monotonically decreasing