"Let $\epsilon > 0$ be given ...." is $\epsilon < 1$ ok?

Question

In limit definitions, is it a requirement that $\epsilon$ is arbitrary, or can I choose to work with small $\epsilon$, say below $1$?

Answer 1

Without loss of generality, you can restrict yourself to $\epsilon \in (0,1)$, or more generally, $\epsilon \in (0,a)$ for each $a > 0$.

To make this formal, you can always modify your argument as follows. You open with, "Let $\epsilon > 0$ be given. Define $\epsilon' = \min(\epsilon,a/2)$." Then you carry out your ordinary argument using $\epsilon'$. Finally, at the end where you conclude $$ ... < \epsilon' $$ you finish with "and $\epsilon' \leq \epsilon$".

Answer 2

$\epsilon$ is arbitrary in definition. For every $\epsilon>0$ there must be a $\delta$ so that the definition holds - it means that definition must be satisfied for every positive $\epsilon$.

If you take for example $\epsilon = 0.5$, consider function:

$f(x)=\begin{cases} 0.1& \text{ if } x \in \mathbb Q \\ 0& \text{ if } x \in \mathbb R\setminus\mathbb Q \end{cases}$

This function has no limit at any $x$ but if we worked with $\epsilon = 0.5$ definition would hold - we could say that it has a limit.