How to say that $\epsilon$ should be small enough in a mathematical statement?

Question

Assuming I approximate $x$ by $y$ throughout minimizing $\|x-y\|_2^2$, I want to define $x$ as the approximate of $y$ when $\|x-y\|_2^2< \epsilon$ in a mathematical definition statement. Then, should I say

"We define $x$ as the approximate of $y$ when $\epsilon$ is sufficiently/arbitrarily/properly small"?

I mean what is the proper mathematical term in such cases to prevent any miss-interpretation and to send a clear message?

Answer 1

Would this work?

For any vector $y\in \mathcal H$ and $\epsilon >0$, the vector $x\in \mathcal H$ is an $\epsilon$-approximation of $y$ if $$ \| x - y\|_2^2 < \epsilon.$$