Proof: Archimedean Principle of Real Numbers | Real Analysis

Given real numbers a and b, where a is positive, we can always find a natural number m so that n*a is greater than b. In other words, we can add a to itself enough times to get a number greater than b. Equivalently, given any real number x, there exists a natural number greater than x, meaning the natural numbers are unbounded above. This is the Archimedean Principle, and we prove it in today's real analysis lesson. #realanalysis

This is a proof by contradiction, making use of the definition of supremum and the completeness axiom/least upper bound property.

Other similar names for this principle: Archimedean Property, Archimede's Principle, Archimedean Axiom, axiom of Archimedes

◉Textbooks I Like◉

★DONATE★

Thanks to Loke Tan, Matt Venia, Micheline, Doug Walker, Odd Hultberg, Marc, Roslyn Goddard, Shlome Ashkenazi, Barbora Sharrock, Mohamad Nossier, Rolf Waefler, Shadow Master, and James Mead for their generous support on Patreon!

Outro music is mine. You cannot find it anywhere, for now.

Follow Wrath of Math on...

[WrathofMath]

Support the production of this course by joining Wrath of Math to access exclusive and early videos, original music, plus the real analysis lecture notes at the premium tier!

[ashutoshsharma]

Great, bro! Last year got much help from your graph theory videos. And now analysis, just amazin'! The crystal clarity in your words is just incomparable and outstanding! I think, that even if the video is not there, audio will be sufficient to clear it.

[aryanshukla]

Thankyou so much, I've watched almost all your real analysis videos and they help so so much ..pls don't stop making these videos ...I am certain more people like me are benefitting from your content ...also since you asked, can you make more real analysis videos, sequence and generally the proofs (writing proofs)

[buseyugruk]

Omg, i was searching all internet for this proof, thank you so much. English isn't even my first number but i understand it.

[teakgrogan]

this is excellent. very precise, clear, easy to follow. 10/10

[antoniodeoliveiranginamaub]

you explain very well, thanks for helping

[aryanshukla]

You took S -1 (by doing so you jumped inside the set N, by JUST ONE STEP), So now when you said, there must be a m in N such that, m>S-1, (well sure, but the set N is discrete and it's not like real no.s, in other words it's not complete) so how can we say that m>S-1 (there is no natural number between S and S-1, so how come you find m ? Is it because S is not Natural no. it's actually a Real no (which is not Natural)?

[johnnelson]

Create a set S belonging to R such that S contains all the elements which don't belong to N. Define m as the least element of N. Then m-1 doesn't belong to N, so it belongs to S. Assume S is inductive, so (m-1) + 1 = m belongs to both S and N (contradiction).

[Davivlcp]

The best youtube channel always saving me

[VEMZOfficial]

Great video but I am just confused: Why MUST m exist? If S-1 isn't a supremum, why must we introduce a value m that is greater than S-1? This is really confusing to me

[valeriereid]

Thanks for this excellent explanation. Really appreciate it.

[aranansivakumar]

Don't usually comment but that was great for my understanding! Great pace too :)

[ConyTrash]

Couldn't you just say:
suppose m is the biggest natural number
=> m+1 is element N because closed under addition and m+1>m, so m can't be the biggest number => there is no biggest number..

[ME-pupp]

Hi sir!. Thank you for the video.
Can you please explain a question for me:

For any  α>0, β>0, prove that there is n∈N such that α/n<β

I notice it is quite similar to the one you've explained.
Thanks in advance

[vedgyan]

Please suggest any book for such clear concept

[MuhammadAsim-mdjg]

Sir please can you tell me effect you used during editing when you write statement like (prove that for any real number x)

[derjemand]

Why do mathematicians make it all so complicated But that was well explained, makes way more sense than my uni lecture notes.

[wizhdan]

You deserve my professor's salary more than himself!

[kantaprasadsinha]

The Archimedean principle ( though Eudoxus principle ) was not specifically stated as real number in most classical text.

[kingraj]

Please i need a solution on this equation. The maximum and minimum of the set
{1-1/3n, n€N}
{1-(-1)^n/n; n€N