 Solution

Put $Z_n=n(1-M_n)$. Its survival function isFor each fixed $z\ge 0$, eventually $n\ge z$, and the exponential limit gives $(1-z/n{)}^n\to e^{-z}$. Thus its cumulative distribution function tends to $0$ for $z<0$ and to $1-e^{-z}$ for $z\ge 0$. This continuous limiting cumulative distribution function identifies the exponential distribution with rate one:This is the exponential endpoint limit of a uniform maximum: the gap from the upper endpoint is typically of order $1/n$, although the maximum itself approaches a constant.