Consistency of an unusual estimator for a binomial parameter

Question

I have a random sample $X_1, \ldots, X_n$ from a $\operatorname{Bi}(N,p)$ population. How do I show that the estimator $$\hat{p} = \frac{\bar{X}}{\max(X_1,\ldots,X_n)}$$ is consistent? The normal way I would approach such a question is to take the expectation and variance of the estimator and use that $\operatorname{MSE}(\hat{p})=\operatorname{Var}(\hat{p}) + \operatorname{Bias}(\hat{p})^2$, but for a ratio of random variables this would be incredibly messy. Is there an easier alternative?

Answer 1

If $Y_n\stackrel{p}{\to}Y$ and $Z_n\stackrel{p}{\to}Z$ then $Y_nZ_n\stackrel{p}{\to}YZ$.

So you might do it by proving that $\overline{X}_n\stackrel{p}{\to}np$ and $\left[\max(X_1,\dots,X_n)\right]^{-1}\stackrel{p}{\to}n^{-1}$.