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 \(x \in D\).