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.
Does mathematics make logical sense? No, it does not. Foundational issues have been finessed by modern mathematicians, and this series aims to turn things around. And it will have interesting things to say also about mathematics education---especially at the primary and high school level. The plan is to start right from the beginning, and to define all the really important concepts of basic mathematics without any waffling or appeals to authority. Roughly we discuss first arithmetic, then geometry, then algebra, then analysis, then set theory. This course is aimed for a general audience, interested in mathematics, or willing to learn.