Let $\left\{X_n\right\}$ be identically, independently exponentially distributed random variables with parameter/mean $1$. Let $M_n$ be the max of the $X_i$'s up to $n$. Find a sequence $\left\{a_n\right\}$ of real numbers such that $P[M_n-a_n\leq x]$ converges to a non-degenerate distribution function $G(x)$ as $n\rightarrow\infty$ for every $x\in\mathbb{R}$ such that $G(x)$ is continuous.
I am using A Probability Path. If this is a well covered subject, can someone direct me to the section that is relevant to the answer?
I know that $P[M_n-a_n\leq X]=\left(1-e^{-(x+a_n)} \right)^n$. I've been trying to manipulate the $a_n$ such that this starts to look like another CDF, but I run into contradictions... so this bashing approach doesn't seem to work. I don't know what concepts this problem is trying to demonstrate. The use of $M$ makes this seem like a Martingale, but I know it's not one since $M_n$ is monotonically nondecreasing.