Let $X_1,...,X_n\sim \text{Uniform}(0,\theta)$. Show that the maximum likelihood estimation (MLE) is consistent.
Setting $Y=\text{max}\{X_1,...,X_n\}$ I know that for any constant $c\in\mathbb{R}$,$$ \mathbb{P}(Y<c)=\mathbb{P}(X_1<c)\mathbb{P}(X_2<c)\cdots \mathbb{P}(X_n<c) $$ but I haven't been able to show consistency yet. Any ideas?