arbitrarily small
‘Arbitrarily small’ does not mean ‘zero’ and it does not mean ‘some fixed tiny number’. It means: name any positive threshold you like, however minuscule, and the quantity can be pushed below it. It is a guarantee about a whole game, not a single move.
Formally, a quantity that depends on a parameter can be made arbitrarily small if for every epsilon > 0 there is a choice of the parameter making the quantity less than epsilon. This is exactly the structure of a limit going to 0: the distance |a_n - L| becomes arbitrarily small as n grows, meaning every positive epsilon is eventually beaten.
The crucial honesty: at no stage is the quantity ever actually zero. ‘Arbitrarily small’ is a statement about the existence of suitable choices for each epsilon, never about reaching a smallest nonzero value (there is none). This is precisely the modern replacement for the old, slippery infinitesimal — a number that was supposed to be smaller than every positive number yet nonzero, which leads to contradiction in the ordinary real numbers.
1/n is arbitrarily small: pick epsilon = 0.001; then for all n > 1000 we have 1/n < 0.001. Pick any smaller epsilon and a larger threshold works. No single 1/n is zero, yet collectively they undercut every positive epsilon.
Smaller than every threshold, yet never actually zero.
Contrast ‘arbitrarily small’ with ‘arbitrarily large’: 1/n is arbitrarily small (beats every epsilon) while n is arbitrarily large (beats every M). Neither sequence ever reaches its target.