Detailed Proof of the Monotone Convergence Theorem | Real Analysis

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We prove a detailed version of the monotone convergence theorem. We'll prove that a monotone sequence converges if and only if it is bounded. In particular, if it is increasing and unbounded, then it diverges to positive infinity, if it is increasing and bounded, then it converges to the supremum of the set of sequence values. If a sequence is decreasing and unbounded, then it diverges to negative infinity, if it is decreasing and bounded then it converges to the infimum of the set of sequence values. #RealAnalysis

Said simply in one case of the theorem: a non decreasing sequence which is bounded above is convergent.

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your intuitive explanations are a force to be reckoned with!

nullspace_xxii.

These days I review my undergraduate mathematics and this video is just perfect for that.... Thank you for clear explanation!

XahhaTheCrimson

Thank you so much for this video, it sure helped me a lot!!!

Your channel is highly underrated and your content is great! Your videos are really good!! I'm glad I found your channel!

_kainya

Your videos help me understand the writings of my real analysis book. You should totally write a text!

vernitasutton

tysm, my book proved the theorem in a really interesting way, but this clarifies it so much!

johnvonneumannsdaddy

Why the assumption of monotonicity in monotone convergence theorem is necessary? Can you give me a detailed answer of this?

lunkadapoorv

A great video!
What drawing app did you use in this video?

AmedK-wblk

Alabado seas!!! Te amoooo, me re ayudaste, Gracias!!!

oyeajugando

We use the completeness property of the reals to conclude that sup(a n) exists when (a n) is increasing, but what do we use to conclude inf(a n) exists when the sequence is decreasing?

FullerHob

i don't understand for the summation of 1/n, where n is an integer from 1 to infinite. While ther series of 1/n to infinite decreases and tends to zero, the Integral test tell us the summation of 1/n is smaller than ln(n) - ln(1) = infinite. However, it fulfills the monotone convergence theoerm conditions, bounded and decreasing. Have I already confused these Mathematic ideas? I'm learning Integral and Series, so please thanks

rayxxkaiser

Bounded does not imply convergent surely? Consider (-1)^n

a.nelprober

Didnt really understand why for increasing sequence the limit should be the supremum

krasimirronkov

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