Show that the MLE divided by the true value converges to 1

Question

Let $S \sim Binomial(N,\theta)$, where $S$ can be seen as $S_n = B_1+...+B_N$ and $B_i$'s are i.d.d $Bernoulli(\theta)$.

I have to show that $\frac{\hat{N}_{MLE}}{N}$ converges to $1$ in probability as $N$ goes to infinity.

I know that $\hat{N}_{MLE}$ is the smallest integer $>\frac{S}{\theta}-1$.

I think I have to use the weak law of large numbers, but I am not sure how to find $E[\hat{N}_{MLE}]$, (maybe it's $E[\frac{S}{\theta}-1]=\frac{n\theta}{\theta}-1$?), or how to find $E[N]$.

Thank you for your help.

Answer 1

By law of large numbers $\frac{S_n}{n}\rightarrow\theta$ in probability.