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> Let $\{X\_n\}\_{n\ge1}$ be a sequence of independent random variables such that $Y\_n=a\_n\sum\limits\_{i=1}^{n}X\_i$ converges in probability to a random variable $Y\_\infty$ for some constants $a\_n\to0$. Show that $Y\_\infty$ is almost surely equal to a constant.
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I have been at this one for some time now with no luck, any help would be greatly appreciated.