Let $(\Omega,\mathbb{F},P)$ be a probability space and $\{X_n\}_{n=1}^{\infty}$ be a sequence of iid random variables such that for each $n \in \mathbb{N}$ and $r \in [1,\infty)$, $P(X_n > r) = r^{-3/2}$. Are there almost surely infinitely many $n$ such that $X_n \in [n^{1/2},n^{4/3}]$?
This is probably an application of Borel Cantelli lemma. At the moment I have that $P(X_n \geq n^{1/2}) = n^{-3/4}$ and $P(X_n \leq n^{4/3}) = 1 -P(X_n \geq n^{4/3}) = 1 - n^{-2}$. I think I must then find a way to write $P(n^{1/2} \leq X_n \leq n^{4/3})$ using these probabilities. Then I can show that $\sum_{n=1}^{\infty}<P(n^{1/2} \leq X_n \leq n^{4/3}) < \infty$ or that $\sum_{n=1}^{\infty}<P(n^{1/2} \leq X_n \leq n^{4/3}) = \infty$ and then apply one of the Borel Cantelli lemmas.