real analysis 101

i did study real analysis at college a decade ago in Vietnam. those days i kept wondering and wondering what the hell it was with stuff like completeness theorem, Cauchy sequence, Riemann integrable insane functions. it didn’t make any sense to me. my instructor made things mysterious as if we, undergrad students, were Newton, Leibniz, Euler, Weierstrass, Riemann sitting in front of him, just like hell.

i couldn’t help thinking of this useless mathematical world. but recently i’ve changed my mind and started doing things from scratch. firstly, it’s a more natural way to track back to the beginning of calculus itself and keep up to its progress onto the modern real analysis we know today. i did some searches and found out a book called The Calculus Gallery: Masterpieces from Newton to Lebesgue by William Dunham and started read it.

it really helps. i can truly understand what Newton and Leibniz did to lay the foundations for calculus as well as Bernoulli brothers and Euler contributions later. i eventually get to know with the ideas that Cauchy, Weierstrass, Riemann created which set the remarkable real analysis era. then i get used to with the dilemmas up to the real number nature that made Cantor, Baire came to the stage.

as my favorite quote of Laozi which reads “a journey of a thousand miles begin with a single step”, calculus should have been introduced as the way it’s really been developed up to date. then it’s not a bloody hell major to students anymore.