I am currently reading through JC Lagarias' "The $3x+1$ Problem and its Generalizations" and have come across some notation reading :

$$\sup_{K \ge 0} T^{(K)}(N)$$

Now I assume that this means "suppose that $K$ is greater than or equal to $0$", however I want to understand any equations I am potentially going to use in a project and thus don't want to rely on assumptions. This may seem a very simple question but I appreciate any help.