Let (left{X_{n}ight}_{n=1}^{infty}) be a sequence of independent random variables where (X_{n}) is a (operatorname{GAmma}(alpha, beta)) random variable

Question:

Let $\left\{X_{n}ight\}_{n=1}^{\infty}$ be a sequence of independent random variables where $X_{n}$ is a $\operatorname{GAmma}(\alpha, \beta)$ random variable with $\alpha=n$ and $\beta=n^{-1}$ for $n \in \mathbb{N}$. Prove that $X_{n} \xrightarrow{p} 1$ as $n ightarrow \infty$.

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