Let (left{X_{n}ight}_{n=1}^{infty}) and (left{Y_{n}ight}_{n=1}^{infty}) be sequences of independent random variables, where (X_{n}) has a (operatorname{Uniform}(0, n)) distribution

Question:

Let $\left\{X_{n}ight\}_{n=1}^{\infty}$ and $\left\{Y_{n}ight\}_{n=1}^{\infty}$ be sequences of independent random variables, where $X_{n}$ has a $\operatorname{Uniform}(0, n)$ distribution and $Y_{n}$ has a $\operatorname{Uniform}\left(0, n^{2}ight)$ distribution for all $n \in \mathbb{N}$. Prove that $X_{n}=o_{p}\left(Y_{n}ight)$ as $n ightarrow \infty$.

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