# Our Daily Story #8: The Rigorous Mathematician with epsilon-delta

http://en.m.wikipedia.org/wiki/Augustin-Louis_Cauchy

We mentioned Augustin Louis Cauchy in the tragic stories of Galois and Abel. Had Cauchy been more generous and kind enough to submit the two young mathematicians’ papers to the French Academy of Sciences, their fates would have been different and they would not have died so young.

Cauchy was excellent in language. He was the 2nd most prolific writer (of Math papers) after Euler in history. When he was a math prodigy, his neighbor — the great French mathematician and scientist Pierre-Simon Laplace — advised Cauchy’s father to focus the boy on language before touching mathematics. (Teachers / Parents take note of the importance of language in Math education.)

Cauchy’s language education made him very rigorous in micro-details. This was the man who developed the most rigorous epsilon-delta Advanced Calculus (called Analysis) after Newton / Lebniz had invented the non-rigorous Calculus (why?).

Rigorous epsilon-delta Analysis:

https://tomcircle.wordpress.com/2013/03/31/432/

The modern epsilon-delta definition by Cauchy was arithmetized by Weierstrass as below:

$\displaystyle\lim_{x\to a}f(x) = L \iff$
$\forall \varepsilon >0, \exists \delta >0$ such that
$\boxed{0<|x-a|<\delta \implies |f(x)-L|< \varepsilon}$

Note:
When I first encountered this epsilon-delta ‘horor’ in the 1st year Math in France, I saw my French classmates reciting at ease the definition learned in their Baccalaureate (but not taught in our British Cambridge G.C.E. A-Level Math):
Quelque soit ε > 0, il existe δ > 0, tel que ...”

If you can’t understand the ‘ε-δ’ definition, just memorize it like a poem – at least better than saying it wrongly:
For all ε > 0, there exists δ > 0, such that …

Watch the excellent Khan Academy Lecture on Limit with ‘ε-δ’ :

(To continue in YouTube on the same webpage of this video)

References:

1. Series Convergence with ‘ε-δ’
https://tomcircle.wordpress.com/2013/05/10/analysis-by-timothy-gowers/

2. Limit with ‘ε-δ’
https://tomcircle.wordpress.com/2013/04/06/limit-%ce%b5-%ce%b4-analysis/

3. Sequence with ‘ε-δ’
https://tomcircle.wordpress.com/2013/05/30/sequence-limit/

4. Limit and Continuity: ‘ε-δ’
https://tomcircle.wordpress.com/2013/04/25/%ce%b5%ce%b4-limit-continuity/