are independent random variables that all have a uniform distribution, U(0,1).

Prove that if then as n approaches infinity, 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

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

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.