Hi I have troubles to understand a proof that is in the the notes and in the book of Terry Tao of Analysis.
I Proposition in question is:
The problems that I have it's to understand some tricky steps. For example to derive $\varepsilon ^2<2$, I only could do it using contradiction; I know, it is kinda obvious but I like to develop every step to be sure of everything. But when I really stuck is where he derives $(2 \varepsilon )^2<2$ and the generalization by induction. So, my question is: could somebody help me to understand this steps?
I would really appreciate some help.
Here is my "sketch" (really is nothing to be honest) of the inductive step:
Let $S$ be the set such that: $n\in S \leftrightarrow n \in N \wedge (n \varepsilon)^2<2$.
Clearly $0\in S$. We suppose that $n\in S$ and we shall show that $n+1 \in S$.
$[\,(n+1) \varepsilon\, ]^2 = (n^2+2n+1)\,\varepsilon^2 ...$
The main thing to recall is that Tao is assuming, to get a contradiction, that $x^2<2$ implies $(x+\varepsilon)^2<2$ for all $x$.
Set $x=0$; one can show $0^2=0<2$, so $\varepsilon^2=(x+\varepsilon)^2<2$ holds. Now, assume $(n\varepsilon)^2<2$. Then set $x=n\varepsilon$, and note that since $x^2<2$, $(x+\varepsilon)^2<2$. That is, $((n+1)\varepsilon)^2<2$.