Definition: A sequence $X_1, X_2, \ldots$ of random variables is said to be uniformly bounded if there exists $M$ such that $\abs{X_n} < M$ for all $X_n$.

Definition: A sequence $f_1, f_2, \ldots$ of functions defined on a domain $D$ is said to be uniformly bounded if there exists $M$ such that $\abs{f_n(x)} < M$ for all $n$ and for all $x \in D$.