Real numbers Cauchy construction

Cauchy Construction of the Real Numbers

In this video, I will show you how to construct the real numbers, but in a cool way! This approach does not use Dedekind cuts or Decimal expansions, but instead Cauchy sequences. Watch this video and be amazed!

[GRBtutorials]

Good timing... right now, we’re about to define the real numbers in my Analysis course! Though now that I think about it, that’s probably the case in most universities in the Northern Hemisphere... so not too much of a coincidence.

[pythoncake]

"let's the problem become the solution"
alright I'll go dissolve myself in acid then :D

[jumperluk]

Makes me happy every time i see it. Thanks

[fellipeparreiras]

Oh man i need to learn Analysis asap to understand this channel!!!

[AnjaniGupta]

It should've started with redefining Cauchy sequences, since the epsilon there can no longer be 'real' that we don't know about as yet. And to redefine, we can just take the epsilon to be any positive non zero rational, as the crux lies in the fact that it still gets arbitrarily close to 0 in the usual metric on Q.

[jimallysonnevado]

In the beginning, we need to emphasize that epsilon is rational because we only know about rationals as of that time and that cauchy criterion is different from the cauchy criterion for reals

[thedoublehelix]

when I defined f(x) to be the special function that solves the ode in my differential equations class, I got points off my test :(

[Kdd]

Dr Peyam is the coolest mathematician 😎😎😎😎

[frozenmoon]

14 seconds in.. what did I notice?

Dr. P. is wearing a Gaussian sweet Integral blouse, he is super happy to introduce us to the Real Numbers, I mean, isn't that just a hot looking "R" on the top-left corner of the board.

[priyanshupareek]

You should definitely consider making more videos

[dgrandlapinblanc]

The cool way... Really ? Whoa !!! Thank you very much.

[aneeshsrinivas]

whoa, you uploaded this 1 day bfore my 18th birthday.

[jkid]

I watched the other videos and thought "these aren't the cool way, it's cauchy sequences"
:) :) :)

[therealAQ]

Metric completion is the way to go! Isn't that the standard to learn about real numbers?

[ritampaul]

Sir, your videos ar really amazing. IA am from India.

[saudinho]

How to show the relation of order does not depend on the representants of the class?

[techtable]

Such a great video! Helped me a lot :))

[elosant]

12:00 Im not entirely sure I understand how the order relation is defined. Is this true for all epsilon > 0? If it were, if we chose two rational numbers r and s then we could choose epsilon > |r-s| and this would surely fail, since both sequences are constant?

[aneeshsrinivas]

why not have the definition of q_n>p_n be that
there exists some number N and some number 𝜖>0 such that when n⩾N q_{n}-p_{n}>𝜖? why does the statement after there exists a number N depend on 2 variables?

[whythosenames]

As you showed that it is ordered you should have shown the axioms i.e. that always [(p_n)] < or = or > [(q_n)]