Consider i.i.d. random variables $X_i$ for $i=1, \dots,n$, where $X_i$ is uniform on $[0,t]$ for $t>0$, which is unkown Consider the esstimator $T_n(x) = \max\{x_1,\dots,x_n\}$, where $x_i$ are the realisations of $X_i$.
For consistency I have to proof that $P_t(|T_n-t|> \epsilon) \rightarrow 0$ for $n \rightarrow \infty \forall t,\epsilon>0$ .
Why can I assume that without loss of generality holds: $0<\epsilon<t$.
Can anybody explain this to me?