Inequalities and more limits | Real numbers and limits Math Foundations 107 | N J Wildberger

The epsilon-delta definition of a limit of a sequence, going back to Cauchy and Weierstrass, is here dramatically simplified by restricting attention to the basic objects of calculus: rational polynumbers (or ``rational functions''). We review the basic definition and give a visual interpretation: instead of an infinite number of nested epsilon neighborhoods, we have a single hyperbolic envelope of the limit. It means that a limit can be certified by exhibiting just two natural numbers: the start m and the scale k.

We illustrate the concept by going carefully over the 3 Exercises from the last video.

Then we begin a quick review/exposition of inequalities, which are important tools when dealing with limits (and analysis more generally). Starting with inequalities for natural numbers, we progressively move to inequalities for integers, and then rational numbers. Notions of positivity are of course crucial here.

Video Content:
00:00 A simpler limit definition
3:09 Re-writing a basic inequality
8:20 Example of a simple limit
12:21 A limit of a rational polynumber
14:04 Inequalities
16:40 Inequalities and how to work with them
20:29 Inequalities with integers
24:36 Positive versus non-negative
26:23 Inequalities with rational numbers
31:27 Further limit exercises

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Here are the Insights into Mathematics Playlists:

[RahulYadav-bpxf]

this is really helpful for me. i am struggling with this for more than a week ....thank you sir

[RENCIOL]

This is so much more clear and LOGICAL then the epsilon delta proofs from first year…..

[MrJosephArthur]

Very interesting efforts of professor Wildberger in order to revolutionize mathematics, maybe understandable in mathematicians with historical mentality. But, discussing about terminological problems, meanings of the terms used in mathematics, etymology of the words would not be necessarily the most important. Analysing the consistency of the new mathematical propositions, especially, if they are well founded, would be much better. Anyway, thinking about a new mathematics is really thrilling.

[crittinger]

Excellent video.

Did you actually have someone else draw the second set of tallies on slide 6?

[jdtaramona]

Dear Professor, thanks a lot for your help and great videos. Now, I would like that you can solve and teach for us, the so called nested radicals, when they become inifnite. I mean radicals of the radicals of the radicals and so on, which is the way to reduce this expression. Thanks in advance.