Let $(X_{n})_{n\geq1}$ be a sequence of independent random variables on a probability space $ (\Omega, \mathcal{F}, \mathbb{P})$ taking non-negative integer values. Suppose that for each $n$ and each $ i \geq 1$, $\mathbb{P}(X_{n} \geq i) = 1/i.$
Let $ M_{n} = max\{X_{k}: 1 \leq k \leq n\} $. I want to show that
$$ lim_{n \to \infty} \frac{logM_{n}}{logn} =1 \, \, \, \, \, \, a.s.$$
almost surely.
I have shown so far that the random variable
$$ limsup_{n \to \infty} \frac{logX_{n}}{logn} = 1 \, \, \, \, \, \, a.s.$$
but I'm not sure how to proceed from here to get the above limit. Many thanks for any help with this!