Definition talk:Sufficiently Small

Regarding the ambiguity referred to in the "also known as" section, what should the $\mathsf{Pr} \infty \mathsf{fWiki}$ convention be regarding using the phrase "sufficiently small" to refer to a negative $\epsilon < 0$ sufficiently small? My vote: use phrases along the lines of "$\epsilon$ for $|\epsilon|$ sufficiently small". This avoids the ambiguity but also feels less awkward than "sufficiently small in absolute value". The same convention could be used for $M \ll 0$ and "$|M|$ sufficiently large". GFauxPas (talk)

"Sufficiently small in magnitude" is adequate if you really want to pin it down. But if you have already specified the $+$ness of $\epsilon$ there's no need to specify its "absolute value" for apparent reasons. If you do ever need to use "sufficiently small" to mean "sufficiently negative", then IMO "sufficiently large (in magnitude)" coupled with a statement to the effect that the quantity in question is specifically less than zero would do the trick. But I don't think I've ever encountered the situation where this has ever been needed.

If you really want to be precise, and $\epsilon$ genuinely might be a positive or negative small value, you can always say "for sufficiently small $|\epsilon|$" and the job is done. --prime mover (talk) 10:42, 28 May 2018 (EDT)

I have always regarded this kind of point as a discussion beyond practical interest, something to keep the students on their toes but which no serious mathematician would bother with. In practically all cases it is abundantly clear what is meant. We ought to spend more time combatting the "exercise to the reader" instead. — Lord_Farin (talk) 12:27, 28 May 2018 (EDT)

You guys have convinced me. --GFauxPas (talk) 13:32, 28 May 2018 (EDT)

By the way, apologies for being abrasive -- I blame a tough day at work. — Lord_Farin (talk) 13:36, 28 May 2018 (EDT)

It's all good, feel better :) --GFauxPas (talk) 13:43, 28 May 2018 (EDT)