Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) and \(\left\{Y_{n}ight\}_{n=1}^{\infty}\) be sequences of independent random variables. Suppose that \(Y_{n}\) is a \(\operatorname{BETA}\left(\alpha_{n}, \beta_{n}ight)\) random variable where \(\left\{\alpha_{n}ight\}_{n=1}^{\infty}\) and \(\left\{\beta_{n}ight\}_{n=1}^{\infty}\) are sequences of positive real numbers that converge to \(\alpha\) and \(\beta\), respectively. Suppose further that, conditional on \(Y_{n}\), the random variable \(X_{n}\) has a \(\operatorname{Binomial}\left(m, Y_{n}ight)\) distribution where \(m\) is a fixed positive integer for all \(n \in \mathbb{N}\). Prove that \(X_{n}=O_{p}(1)\) as \(n ightarrow \infty\).