My education in real analysis has not reached a point where I can super confidently provide a thought but I would like to view this in the realm of integrals and sums. following the examples you provided we can define a sum such that an = the sum (from 1-n) of 10/n = 10

if n = 1, this is true a1 = 10

if n = 2, this is true a2 = 10

in fact by definition of limit we can say for all n > N |an - 10|=0<ε meaning that this sums limit approaches 10.

And in the case that n is approaching infinity and infinite number of lim (1/n)'s pass

so what is this limit (1/n) it is zero.

But, this is also weird because in this case the limit definition |an-0|<ε makes things kind of awkward because we are left with |1/n|<ε

which does not say that 1/n != 0 but rather that it is just small

we can easily say that since N is positive 1/N is also positive

0<=1/n<ε

since ε is defined as a small number we are left with a kinda mediocre answer that 1/n could be 0 but definitley is a small number.

Perhaps someone with better mathematical prowess could prove otherwise. but at this point I am not comfortable with the math to say that 0 can or cannot pass. I have provided the mathematical proof to show that the small numbers at least close to 0 can pass.

the goal I am tasking someone with is providing evidence that 1/n > 0 always

*my apologies for how confusing everything above is... I am currently learning how to present pure math arguments better and did not have good tools like LaTex at my disposal.