Monotone Sequence Theorem

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In this video, I prove the monotone sequence theorem in calculus, which says that if a sequence is increasing and bounded above, then it must converge. It’s a great way of showing that a sequence converges, without actually finding the limit!

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Dr. Peyam have you considered giving an Analysis in R^n course or perhaps making videos in this area with examples and some of the most important results and their proofs? It would be a great help. Thanks for all your videos.

camiloc

From your last example, it seems the monotone theorem applies to the infimum of a monotonically decreasing function as well. Is this correct?

historybuff

My man! thanks for this although we call it the monotone convergence theorem :P

raichuk

Disappointed that this lecture was not voiced at constant pitch.

rogerkearns

Wow very nice 👍🏼

Will you prove the Weierstraß teorem as well.

Myrslokstok

Potentially controversial but what is your opinion of constructivists or finitists who deny the existence of real numbers? People like NJ Wildberger or Doron Zeilberger? Personally, I think you can build a finite mathematical system but you're just making the effort much more harder than it needs to be if you accept infinities.

theproofessayist

I need you as professor for analysis lol

Maniclout

1:11 'universal constant c' makes me think of speed.

gordonchan

This theorem can be viewed as an axiom, since the existence of supp in the real numbers for a bounded set is an AXIOM of the real numbers. So, if you believe in the UPPER BOUND PROPERTY of "real numbers" then this "theorem" is true. There's not a constructive model for real numbers (yet).

FranciscoSeoane

what do you think about this book? Curso de Análise - Elon Lages Lima (isnt the book Análise Real of the same author)

Lklibertad

So can I assume the monotone decreasing function 1/n limit exists and (and approaches 0), but can the monotone increasing function sum 1/n n->infinity function can be proved to be unbounded by using this theorem?

dhunt

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