Proof: Supremum of {n/(n+1)} = 1 | Real Analysis

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Today we prove the supremum of {n/(n+1)} is 1, using the Archimedean principle and the epsilon definition of supremum of a set.

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!

+WRATH OF MATH+

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wow this was really awesome, im just starting out with real analysis and i found your general method of proofs to be super helpful!

filip

Thank you for your explanation. Really helpful.

pakwidi

yeah im gonna accept that im gonna fail

Jusexle

when proving that for all epsilon>0 (e>0), 1-e is not an upper bound. would this work: we could fix e>0 but if e>1 then we just pick 1/2 which is in S to show an example of an element in S greater than 1-e, so we just fix 0<e_<1, then for any epsilon in this interval we pick x = [ceiling(1-e/e) +1 / ceiling(1-e/e) +2] which would be in S and also greater than 1-e to show an example of an element is S greater than 1-e?

parthmahind

I wish my professor went over the details of proofs like this but he wanted to just show the definitions and let us explore as if we were pursuing a Ph.D in math. I almost tricked him and got him to give a solution to an exam problem one time lol. Anyway, great explanation.

GETURHANDSUP

That's really well explained and carried me through the first week of college

dominiorrr

thanks a lot. I messed up a real analysis quiz cuz i found the topic quite boring but now I discovered your channel and have a newfound interest in it. Might ace it this semester

aryansudan

Wow. This is amazing. Thank you so much!!!

oxman

Wow this makes proving them so simple! I had an assignment question for a proof like this and prove it assuming a set had a maximum (which not all sets do 😅). These would've been so helpful, Definitely taking this into my final exam ♥

blakebodycote

Thanks for the video : )
a) Can you say that S = {0.5, 2/3, 0.75....}, so the elements of S form an increasing sequence where lim as n-->infinity(1 + 1/n) = 1. So x<1 --> x<=1, so 1 is an upper bound of S
b) Could you say that given any epsilon > 0, 1 - epsilon > 1- 0.5epsilon which is an element of S, so 1 is the smallest upper bound of S. I don't think this is right though because we are only considering natural numbers N rather than the entire set of real numbers

sportmaster

How would you go about this problem when you have to find the sup/inf, not prove that a given one is indeed true? I have to find inf/sup of 2-(1/(n+1)) and i know that inf= 3/2 but i dont think I'm allowed to state this at the beginning of my proof since its not given. A reply would be very nice :)

michieldolfing

That first part of the explanation is so intuitive. Hope you make more on these advanced topics.

nonentity

as always great video! i was just wondering, can I start the proof for second condition by choosing epsilon > 1/n+1 ? I have other problems but it couldn’t be broken down into n < …

nobirapals

hi
I am unable to understand your conclusion for the second part of the proof. Can you help me out?

IamBrill

am I right we could start that prooj from Archimedean principle and that would be clear one ? I mean no need for the first part except just to understand where it come from

dandan

also it is a cool thing how we have different perception of the math topics. I mean i have different school but we never did such approach to that topic. Cool one

dandan

Somebody please ans me - glb and lub of mn/m^2 + n^2, m belongs to reals and n belongs to natural number.

tania

Wow, cool sir am happy knowing this channel. Am new in Analysis. I wish we will ride together😅.

Watching from Africa❤

AFCOE

Hey, yes I agree- you may be the best teacher I've found yet and I'm just saying-that's saying a lot. Even though, I don't understand any of it, there is hope that one day I might. In the meantime, could you please show a proof for what the sup and inf for e^x are? It seems like a basic question/proof but, it would be an amazing video I'm sure. Unless you already have one, but if you do- I can't find it. I love your videos. Keep going! You are a blessing to those of us who need things broken down so much.

KristineSteele-fsyc

hey do you know how to prove this without any use of limits, as in just by showing that any real number x with x < 1 is not an
upper bound, by finding an element of S larger than x?

jasmineb

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