Showing that a max likelihood estimator is unique.

I'm given that $\displaystyle f(y;\lambda) = \lambda(1-y)^{\lambda-1}$

where $\displaystyle 0 < y < 1$ and $\displaystyle \lambda > 0$

I've worked out that the max likelihood est of $\displaystyle \lambda$ based on $\displaystyle Y_{1},...,Y_{N}$ is

$\displaystyle \lambda_{ML} = - \frac{n}{\sum log(1-y_{i})}$

Firstly, is this correct?

Secondly, how would I show that it's unique? Am I right in thinking that if it's unique, as n gets bigger, $\displaystyle \lambda_{ML}$ tends to $\displaystyle \lambda$? If so, how do I show this?