Past exam of the probability course of the University of Oxford / 2015 / 1 / a / i / Solution

The sample maximum has cumulative distribution functionThe product $x^n$ follows from independence of random variables and the continuous uniform distribution of each observation. For every $x<1$, this cumulative distribution function tends to zero; for $x>1$ it equals one. These are exactly the values of the cumulative distribution function of the constant random variable $1$ at its continuity points. Therefore, by the definition of convergence in distribution,At $x=1$ the limiting cumulative distribution function is discontinuous, so the definition does not require checking that point, although $F_{M_n}(1)=1$ also holds.