How I convince my self that if we say $\epsilon >0$ we must refer to small quantity?

Question

Many mathematical definitions almost used $\epsilon >0$ to define any mathematical notion, for example if we want to give definition to the convergence of sequence we say " $\forall \epsilon >0 , \cdots $ which means $\epsilon \in (0,\infty)$ than $\epsilon$ can take any large value since its belong to this domain $(0,\infty)$ , I have read here that late mathematician P. Erdős also used the term "epsilons" to refer to children (Hoffman 1998, p. 4). But how I can convince my self that epsilon must refer to small quantity since we assumed it greater than $0$ ?

Answer 1

In a typical analysis context, the statement to be proved is of the form

For all $\epsilon > 0$ there exists ... such that $ \ldots < \epsilon$

There is no need for $\epsilon$ to be small, but if the statement is true for some $\epsilon_0$, then it is automatically true for all larger $\epsilon$. So the hard part is to deal with small $\epsilon$.

Answer 2

You cannot, and really must not. If you are given that $\varepsilon>0$, that's really all you can assume about it. For instance, consider the following start of an argument:

Let $\varepsilon>0$ be given. Let $\delta:=\varepsilon^2$. Since $\delta<\varepsilon$,...

Well, no. Even though this is a delta-epsilon proof and your attention is focused on small values of epsilon, the statement is not true if $\varepsilon>1$. In cases like this, you must take $\delta:=\min\{.1,\varepsilon^2\}$ or something similar to have a valid argument.