$\displaystyle X_1, X_2,..... $ are independent random variables that all have a uniform distribution, U(0,1).

Prove that if $\displaystyle M_n = max (X_1, X_2, ..., X_n) $ then as n approaches infinity, $\displaystyle n(1-M_n) $ converges in distribution to an Exp(1) random variable.

I've done a little bit but I have confused myself and I don't know what to do with it now (and im not even sure if Im on the right track ).

So

$\displaystyle Pr(n(1-M_n) \leq x) = Pr(1 - \frac{x}{n} \leq M_n)$

$\displaystyle = 1 - Pr(M_n < 1 - \frac{x}{n})$

$\displaystyle = 1 - Pr(\forall n, X_n < 1 - \frac{x}{n}) $

$\displaystyle = 1 - \prod_{i=1}^n \ Pr(X_i < 1 - \frac{x}{n})$

$\displaystyle = 1 - (1-\frac{x}{n})^n$

and this is where I get lost. I think maybe.....

$\displaystyle = 1 - exp(n \ln (1 - \frac{x}{n}))$

but then where to go from here I dont know, which makes me doubt the accuracy of what I have done. Thanks for any help you can provide me with.