(Maximum of uniforms) if $X_1,X_2,\ldots$ are iid uniform $(0, 1)$ and $X_{(n)}=max_1\leq i\leq X_i$, let us examine if (and to where) $\;X_{(n)}$ converges in distribution. As $n \implies \infty$, we expect $\;X_{(n)}$ to get close to 1 and, as $\;X_{(n)}$ must be less than 1, we have for any $\epsilon > 0$,
etc...
The thing i don't get is "as $\;X_{(n)}$ must be less than 1". I thought as uniform continuos variable the sample space is defined in between $\{a\leq x\leq b\}$ so in this case it would be $\{0\leq x\leq 1\}$ and the statement should be "as () must be less or equal than 1". Is a typo or I'm just wrong?
In any case, it will not chance anything, since $\mathbb P\left(X_{(n)}=1\right)=0$.