I'm struggling with the following problem: Suppose there are $n$ bins and balls arrive as a Poisson Process with rate 1. On arrival each ball falls into a bin uniformly likely. Let $M_n$ be the maximum number of balls in any bin at time $n$. Show that
$$P\left(M_n\ge(1+\epsilon)\frac{\log{n}}{\log \log{n}}\right) \to0$$ with $n$.
And we are given the 'hint' that if $X$ a Poisson Process with rate $1$, then $$P(X\ge x)\le \exp(x-x\log{x})$$
My idea was to use the fact that since #balls in each bin is independent; $P(M_n<k) = P($all bins have $\le k$ members) = $P$(bin 1 has $\le k$ members)$^n$. And as arrivals to bin $1$ are Poisson with rate $1/n$, this is equal to $$\left(e^{-1}(1+1+1/{2!}+...+1/{n!})\right)^n$$ which we want to show converges to $1$. But this is really turning the problem into an analysis question rather than a probability question which is not what I want to do. Also it doesn't use the hint.