What does $\sup _n$ mean?

Question

I came across this notation when studying about the Kolmogorov Convergence Criterion, namely

Suppose that $\sup _n |X_n| \leq A$ for some $A>0$, ...

I understand what does $\sup$ mean, but what does $\sup_n$ mean?

I have tried to search for more information online but did not manage to find anything.

Answer 1

It means the supremum of the possible values of $|X_n|$ as you vary $n$ (presumably in the domain of $X_n$).

Answer 2

If $(a_n)$ is a sequence of real numbers, which is bounded from above, then let $A:=\{a_1,a_2,a_3,...\}$.

By definition we have

$$\sup_n a_n= \sup A.$$