Proof: Infimum of {1/n} = 0 | Real Analysis

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The infimum of the set containing all reciprocals of natural numbers has an infimum of 0. That is, 0 is the greatest lower bound of {1/n | n is natural}. We prove this infimum in today's real analysis lesson using the Archimedean Principle, which tells us that given any real number x, we can find a greater natural number.

Thanks to Nasser Alhouti, Robert Rennie, Barbara Sharrock, and Lyndon for their generous support on Patreon!

I hope you find this video helpful, and be sure to ask any questions down in the comments!

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preparing for my exam and it's the first part of what we learned this term, seems my mind is blank... Thank you for your video!

annali

Wonders my situation if these lectures weren't there

antebantesaathmaintante

I think that if we want to divide by n both sides in the inequality 0<=1 we must first assure that n is not equal to zero, as zero belongs to N set.

ioannis

2:54 You forgot to mention that n must be greater or equal to 0
n can't be negative

RazorM

1/Epsilon does not belong to the set of real number since it doesn’t include 0 and the negatives so that mean the archamdian property does not apply. Right?

Rayglobster

thank you for these videos I am 9th a 9th grader who really wants to learn real analysis but the problem is just reading the book dosen't help me understand as much as these videos do also any suggestions on how I could self-teach myself such kinds of stuff where I cannot just sit read the book and understand what's going on

adityaekbote

My favorite part is when you use the first order symbols! I've never used them so "Backwards E, x, curved E, R with an extra line, such that, x >= n, upside down A, n, curved E, N with an extra line..." sounds like a new language! How did you go about learning to use these symbols?

cheatyhotbeef

You can also just start with n>0 because n is a natural number. The lowest natural number is 1, and obviously 1>0.
So (1/n)>0, dividing both sides by n. This shows 0 is a lower bound.

garyanderson

S={sin (nπ/6 ;n belong to natural number}.sir tell me suprimum

manahilfatima

Hey I just wanna say all this stuff is really fucking useful for students! I’m watching this to cram for my proofs final in a few hours, and tysm for this work

rbran

Why that o is a lower bound of the set??

eddiemoring

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