Is ##\delta##-steady needed in this proof, given ##\epsilon##-steady

[yucheng]

TL;DR Summary

Is ##\delta##-steady needed in this proof, given ##\epsilon##-steady for all ##\epsilon>0##?

In Tao's Analysis 1, Lemma 5.3.6, he claims that "We know that ##(a_n)_{n=1}^{\infty}## is eventually ##\delta##-steady for everyvalue of ##\delta>0##. This implies that it is not only ##\epsilon##-steady, ##\forall\epsilon>0##, but also ##\epsilon/ 2##-steady."

My question is, why do we need the statement on ##\delta## when we already have ##\epsilon##-steady, ##\forall\epsilon>0##, which immediately follows that the sequence is ##\epsilon /2##-steady since ##\epsilon>0 \implies \epsilon /2>0##? Is this just his style, or is it logically necessary?

 

[mfb]

when we already have ##\epsilon##-steady

Don't you get that from ##\delta##? I don't have the book so maybe the context is relevant, but you need to start with some knowledge to conclude anything.

 

[yucheng]

Don't you get that from ##\delta##? I don't have the book so maybe the context is relevant, but you need to start with some knowledge to conclude anything.

Clarification: the 'knowledge' is ##\forall\epsilon>0##... For whatever reason, the author used ##\delta>0##... in the proof instead, then only brought ##\epsilon## in later.

Is it to distance the ##\epsilon##, I mean the definition for a Cauchy sequence already uses ##\epsilon##, so if I want to say ##\epsilon /2## fulfils the condition, I can reframe the definition in terms of ##\delta >0##, then point out that ##\epsilon >0## also fulfils the condition, i.e. (##\epsilon\in \{x:x = \delta\}##) it, then ##\epsilon /2>0## also fulfils it, i.e. ##\exists\epsilon /2: \epsilon /2\in \{x:x = \epsilon\} \subset \{x:x = \delta\}##?

I apologize if I am abusing, if not misusing notation!