= Solution
For $x>0$, the event $M_n/n\leq x$ says that each of the <independent random variables> is at most $\lfloor nx\rfloor$. Hence, for all sufficiently large $n$,
$$
\mathbb P(M_n/n\leq x)=\left(1-\frac1{\lfloor nx\rfloor+1}\right)^n.
$$
Since $n/(\lfloor nx\rfloor+1)\to1/x$ and $\log(1-u)=-u+o(u)$ as $u\to0$, taking <logarithms> gives the <limit> $e^{-1/x}$. For $x\leq0$ the <probability> is zero. Therefore
$$
\boxed{\frac{M_n}{n}\xrightarrow{d}W,\qquad F_W(x)=\begin{cases}0,&x\leq0,\\ e^{-1/x},&x>0.\end{cases}}
$$
This is the <Fréchet limit for a reciprocal integer tail>, with a <Fréchet distribution> of shape one. Its <cumulative distribution function> is continuous even at zero, so the pointwise calculation establishes <convergence in distribution>.
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