Monotone Sequence Theorem

Monotone Sequence Theorem

In this video, I prove the celebrated Monotone Sequence Theorem, which says that an increasing sequence that is bounded above must converge. This is the fundamental theorem of sequences, because it allows us to easily prove convergence of sequences

[beatoriche]

I really like your tendency to draw meaningful pictures. A common pattern of behavior I see from high school students to math majors in university is not drawing pictures, and I honestly don't understand why, since not thinking visually just makes everything so much more difficult. I've recently been reading a measure theory textbook, and a lot of the steps are quite difficult to follow on their own (some definitions in particular seem completely arbitrary), but they just pop out if you draw a nice picture. I really think it's difficult to overstate just how valuable visual intuition is when it comes to understanding abstract things; some people seem to believe being rigorous entails not thinking visually, but I really don't get this dichotomy – it's just important to keep in mind that pictures are merely aids in our thinking and that we have to make real arguments instead of just pointing at the picture.

[punditgi]

An elegant explanation of an elegant theorem!

[tomesval]

Want To Find (WTF), well played Dr. Peyam...

[plaustrarius]

Professor, I recently stumbled upon a sequence i find pretty cool and haven't really seen or heard anyone talk about


Aronson's sequence is an integer sequence defined by the English sentence "T is the first, fourth, eleventh, sixteenth, ... letter in this sentence."

That's so wild to me! Hope you enjoy!

Thank you!

[VanDerHaegenTheStampede]

Dear Dr Peyam. The Mean Value Theorem states that if f(x) is a differentiable function, then between any two values x = a and x = b there is a value x = c such that f'(c) = average rate of change. That is, the tangent at f(c) is parallel to the line through points f(a) and f(b).
It is worth noting that:
i) For f(x) = x^2, c = (a + b)/2, that is, the arithmetic mean;
ii) For f(x) = 1/x, c = sqrt(a b), the geometric mean;
iii) But, for what function f(x) would c = (2 a b)/(a + b), the harmonic mean?
I couldn't follow the book* by Sahoo and Ridel, which by the way relate f(x) = ln(x) to the identric mean, beyond other generalized means. So I humbly request your help for (iii). Thanks in advance.
(*) "Mean Value Theorems and Functional Equations" by P. K. Sahoo, T. Riedel.

[lowlifeuk]

Dr Peyam, you have done recently a lot of works on limits. I personally am very confused about hyperreal numbers and how they got postulated and more importantly why they came object of study. I personally refuse the notion of hyperreal number, but of course if serious mathematicians accept and use them, there must be something. I would be very grateful if you could shed some light on the subject. Thank you.

[yousifkamaash]

Thanks for this lecture. I have a question in geometry . What are the dimensions of an equal sides Pentagon (upside down ) inscribed inside a triangle whose sides dimensions are 77, 106, 113 . Knowing that the individual angles of Pentagon measures 108 degree and the summation of angles are 540 degrees. with my best regards .

[sumitgupta]

just one question at 7:40 how do we ensure that s- € is in the set S of all elements of sequence? As for the definition of the sup to work the element smaller than should also belong to the set of whose sup we are considering.

[MrCigarro]

Professor, I was given the following sequence: 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 21, 25, 26, 29, 32, and I was told that the next 5 points were: 33, 34, 36, 37, 40. I was unable to make sense of this. Could someone make sense of it?

[sunilkumargupta]

Professor...I face slightly difficulty in definite integration. Can u help me by some videos ? (Full definite integration) it will be really helpful. Love from 🇮🇳

[GreenMeansGOF]

Corollary: If a_n is monotone decreasing and b_n is monotone increasing and a_n ≥ b_n, then each sequence converges.

Proof: a_1 ≥ a_n ≥ b_n ≥ b_1 and thus each sequence is bounded and convergent.

[pierreabbat]

If Modern Greek is bounded, then it converges. Ancient Greek, we're not so sure. A circumflex sequence might wander between bounds without converging.

[adityachk]

Hey what happened to Steve from black pen red pen?

[thedoublehelix]

This would also work is a(n-1)<=an (or >=) for all n. Also, I know you pre recorded these videos but it would be cool if you made a video proving that every sequence has a monotone subsequence. Theres a really cool proof involving peaks for that fact. Then you get Bolzano–Weierstrass theorem for free!

[pbj]

Dr. Peyam, you should make some fun videos once in a while as well. While Real Analysis is good, most of the time we're proving simple results which are intuitive anyway

Love your attitude towards math 😄👍👍

[vikramchoudhary]

I'm indian🇮🇳🇮🇳🇮🇳🇮🇳🇮🇳🇮🇳🇮🇳🇮🇳🇮🇳 so this theorem is very important in bsc sem.. 2

[juanmanuelmillansanchez]

Dr it was an amazing proof video, but i have a question remaining if it is possible for you to answer it . ¿ How do I find the bounds ? and why should i find them to see if a sequence is convergent besides just taking the limit?.thanks for the content.

[anirvinvaddiyar]

1 = sqrt(1)
sqrt(1) = sqrt(-1 x -1)
sqrt(-1 x -1) = sqrt(-1) x sqrt(-1)
= i x i = -1

1 = -1?

line 3 is the error but can someone plz why?

[watchaccount]

Do you know about Aspergers' Syndrome?

[ss-kpny]

Sir vedio uplode green function visualisation graphically reply