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From this question, I wonder if the following could be proved (this is probably a Borel-Cantelli trick):

Let $X_n$ be a family of independent random variables with a common density $f$ with respect to the restriction of the Lebesgue measure on $[0,1]$.

In other words, the law of $X_1 = X$ is $$ P(X\in B) = \int_{[0,1]\cap B} f(x) dx. $$ Let us define $Y_n = X_n^n$. Then $$ Y_n \to 0 \ \ \ \text{a.s.} $$

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No, quite the opposite. For example, consider the uniform distribution. Then for any $b \in (0,1)$, $$P(Y_n > b) = P(X_n > b^{1/n}) = 1-b^{1/n} \approx \dfrac{1}{n} \ln(1/b) \ \text{as}\ n \to \infty$$ Since $\sum_n 1/n = \infty$, Borel-Cantelli says $Y_n > b$ infinitely often.

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