Definition:Sufficiently Small

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Definition

Let $P$ be a property of real numbers.

Then:

$P \left({x}\right)$ holds for all sufficiently small $x$

if and only if:

$\exists \epsilon \in \R: \forall x \in \R: \left\lvert{x}\right\rvert \le \epsilon: P \left({x}\right)$

That is, if and only if:

There exists a real number $\epsilon$ such that for every (real) number not more than $\epsilon$ in in absolute value, the property $P$ holds.


It is not necessarily the case, for a given property $P$ about which such a statement is made, that the value of $\epsilon$ actually needs to be known, merely that such a value can be demonstrated to exist.


Also see