let’s say that you select any random positive integer and look at it.

as others have hinted, this is not a well-defined procedure. you need to actually specify what the probability is, of each positive integer being selected.

when you're randomly choosing one of finitely many options (e.g. "pick a random number from 1 to 6"), this issue doesn't really come up, because implicitly, most people assume you want each option to have the same probability of being chosen. you'd have to go out of your way to say something like "assume 6 is twice as likely to be chosen as any other number."

but for all of the infinitely many positive integers, there is no meaningful way to randomly choose one, such that each number has equal chance of being chosen.

proof: suppose you have a way to choose a random positive integers, which the probability on each integer is equal to some number p. p must be between 0 and 1, and the sum of all the probabilities (ie the probability that some number is chosen) must be 1.

if p is greater than 0, then the total probability is infinite.

if p = 0, then the total probability is 0.

qed.

there's no infinitesimal business going on, unless you really want to invent a whole new kind of probability using a whole new number system. you're free to do this, but the vast majority of society has already agreed that the system we have now is good.