What is the meaning of 'sup' in this equation

Question

What is the meaning of 'sup' in this equation: $$s(n)= \frac1n\sup_{k\ge0}T^{(k)}(n)$$

The equation is from this paper on the Collatz conjecture.

Answer 1

Consider the set $\{T^{(k)}(n):k=0,1,2,\ldots\}.$ Then, $\sup_{k\ge0}T^{(k)}(n)$ is exactly $\sup\{T^{(k)}(n):k=0,1,2,\ldots\}.$ The supremum $\sup A$, where $A$ is a subset of an ordered set $E$, is an element of $E$ such that $x\le \sup A$ for all $x\in A$ and if $b < \sup A$ then there is some $x\in A$ such that $b < x$.

In other words, $\sup A$ is an upper bound of $A$ and it is the smallest upper bound of $A$.