User:IssaRice/Formalizing adjustment of epsilons: Difference between revisions

Revision as of 06:13, 24 May 2020

I want to elaborate on a thing that shows up when proving part (a) of https://taoanalysis.wordpress.com/2020/05/24/exercise-5-6-1/

Actually, a similar thing appears in Tao's original proof (Proposition 5.5.12) and the situation is simpler there (fewer variables to deal with), so let's start there.

Tao's proof

In the first part of the proof, we suppose $x^{2} < 2$ , and let $0 < ε < 1$ . Then we show the bound $(x + ε)^{2} \leq x^{2} + 5 ε$ . Now we say that since $x^{2} < 2$ , we can choose $0 < ε < 1$ small enough that $x^{2} + 5 ε < 2$ .

@@ Line 5: / Line 5: @@
 ==Tao's proof==
-In the first part of the proof, we suppose <math>x^2 < 2</math>, and let <math>0 < \varepsilon < 1</math>. Then we show the bound <math>(x+\varepsilon)^2 \leq x^2 + 5\varepsilon</math>.
+In the first part of the proof, we suppose <math>x^2 < 2</math>, and let <math>0 < \varepsilon < 1</math>. Then we show the bound <math>(x+\varepsilon)^2 \leq x^2 + 5\varepsilon</math>. Now we say that since <math>x^2 < 2</math>, we can choose <math>0 < \varepsilon < 1</math> small enough that <math>x^2 + 5\varepsilon < 2</math>.
 ==The general case==