Proof: Archimedean Principle of Real Numbers | Real Analysis
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- čas přidán 20. 08. 2024
- 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.
Definition of Supremum and Infimum: • Definition of Supremum...
Real Analysis Course: • Real Analysis
Real Analysis exercises: • Real Analysis Exercises
Other similar names for this principle: Archimedean Property, Archimede's Principle, Archimedean Axiom, axiom of Archimedes
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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)
You're very welcome and thanks a lot for watching, Aryan! So glad they have been helpful and more are on the way. I'll try to do more on sequences soon!
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.
Thanks so much Ashutosh! I am glad to hear you've found the lessons helpful and I hope you'll continue to find the many upcoming real analysis lessons helpful!
Omg, i was searching all internet for this proof, thank you so much. English isn't even my first number but i understand it.
this is excellent. very precise, clear, easy to follow. 10/10
So glad to hear it! Thanks for watching and if you're looking for more analysis check out my playlist! czcams.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html
Many thanks for this good video.
you explain very well, thanks for helping
Thanks Antonio, glad to help! If you're looking for more analysis, check out my Real Analysis playlist and let me know if you ever have any video requests! czcams.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html
Don't usually comment but that was great for my understanding! Great pace too :)
Why do mathematicians make it all so complicated lolllll. But that was well explained, makes way more sense than my uni lecture notes.
Haha, the most basic stuff can be very complicated in that we have to be exceptionally careful about what we are assuming! Thanks for watching and let me know if you ever have any questions; if you're looking for more analysis, check out my playlist! czcams.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html
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).
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..
Good question. Supposing there is a biggest natural number is a stronger assumption than what we did in the proof. In the proof we only assumed the naturals were bounded by some real number. Thus, we worked with a supremum which wasn't necessarily a natural. Additionally, proving there is no biggest natural number doesn't necessarily prove that given a real x, there is a natural greater than x. For example, there is no biggest member of the set S = { n/(n+1) }, but we cannot find an element of S greater than any given x>=1.
@@WrathofMath Cheers for clearing that up!^^
The best youtube channel always saving me
Thanks so much! I'm glad to help!
Thanks for this excellent explanation. Really appreciate it.
Glad to help - thanks for watching!
Thank you so much sir
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)?
I'm not sure if I understand your question, but let me try to clear it up! We assume, for contradiction, that N has an upper bound. Thus, it is a bounded and nonempty subset of the reals, which are complete, and so N must have a supremum. We let sup N = S. Thus, since S is the least upper bound of N by definition, we know that S - 1 is not an upper bound of N, and so N must contain some m > S - 1. Hence, m + 1 > S. But N is closed under addition, and so m + 1 is in N, contradicting S being the supremum, and thus N has no upper bound.
Like you said, S is not necessarily a natural, but since it is sup N, we immediately have by definition that S - 1 can't be an upper bound of N. By definition of S-1 "not being an upper bound", N must contain an element greater than S-1.
Does that help?
@@WrathofMath very much , I also contemplated a lot on this but couldn't get the detail but when I did I was not very sure , but after reading your last paragraph , things became crystal clear. Thankyou so much
@@WrathofMath Hii. I read your explanation but I'm still a tad confused. I understand the method but just thinking about it logically, if sup N = S and m > S-1, then isn't it logical to assume that m = S. If S is a natural number, then there are no more natural numbers in between S and S-1.
I understand the math behind it but I can't quite grasp the logic behind it. Please clarify if you can :)
I agree with you, but not sure I understand this proof well enough. @@anaghavarma751
why m+1 isn't equal to S?
The Archimedean principle ( though Eudoxus principle ) was not specifically stated as real number in most classical text.
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
Your question is 6 months old, but if it helps, here's the answer. The supremum is not in N, it's a real number. So if you take a real number s and subtract one from it, you'll find an integer n that's greater than s-1. I think the confusion comes from thinking that sup(N) is a natural number when it's not.
Suppose if I take a as 6 and b as 20 as there is no condition on b then 3a is less than b.
A very descriptive explanation.
Thanks for watching!
You deserve my professor's salary more than himself!
Haha, I do my best! Send him my PayPal donation link www.paypal.com/paypalme/wrathofmath and let him know! Be sure to check out the real analysis playlist if you're studying the subject, lots more lessons are on the way! czcams.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html
@@WrathofMath Thanks alot 😊
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
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
Take what we've done here: n*a > b, and switch the roles of a and b, the n*b> a, or b>a/n.
Please suggest any book for such clear concept
Thank you !!
Happy to help!
I have a doubt sir
nx greater than y
It is not possible for all real numbers
Suppose we take n=2 and a = 1.5 and b=10.5 then it is not satisfy the archimedian property ..so its a confusing
the point is that there will be some n that will satisfy the property not that all n will satisfy the property. So in your example n = 1000 will satisfy the property.
too good...better than the books atleast
Thanks a lot, glad it helped! If you're looking for more analysis, check out my analysis playlist: czcams.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html
What about equality
prove: Archimedean principle use proof by contradiction and completeness axiom ❣️ please prove this thank you
Thanks for watching! Did you watch this lesson? That's precisely the strategy we used! Let me know if you have any questions about it.
Thanks for the great video. My problem with this axiom is that I can manipulate this proof in a way that there is always a real number bigger then a natural number ( like x*m>n with x is real and m,n is natural). Can anyone help? I am new to prooving things :D
Sir please can you tell me effect you used during editing when you write statement like (prove that for any real number x)
Thanks for watching, that is just a simple fade in. Any video editor you use should be able to do that. I'm not fading the statement in, I'm simply fading from one clip without the statement to a clip where the statement has already been written, so it looks like just the statement is fading in since everything else is the same. And if you're looking for more analysis, check out my playlist! czcams.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html
@@WrathofMath Thanks a lot brilliant Sir From pakistan❤
Great work...
You were speaking too fast in this video
I think you should take your time to speak so we can hear you well and understand the concept well.
Thank you! I always try to speak very clearly and not too fast, and I don't get many complaints about my speed - the ones I do get are a mix between "too fast" and "too slow", it's impossible to be just right for everybody, but I just try to be as clear as possible! If you find a particular lesson too fast, you can always try watching it again, or watching certain sections of it again. There is also often additional explanation in the description, as well as links to related lessons you may find useful. And if you have any questions feel free to ask. Sometimes what you need to do is just grab pencil and paper and write out some of the stuff yourself, that's often the best way to iron out the details you're struggling with!
He spoke just fine just learn proper English first or slow down the video.
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