Past exam of the options I course of the University of Oxford / 2013 / M / 2

For $0\le z\le 1$, independence and the uniform distribution giveSince $M_n\le 1$, for $0<\epsilon <1$,For $\epsilon \ge 1$ the probability is zero. This is exactly convergence in probability of $M_n$ to one.

Put $Y_n=n(1-M_n)\ge 0$. For fixed $x\ge 0$ and sufficiently large $n>x$,For $x<0$ its distribution function is zero. The limiting distribution function is therefore $F(x)=0$ for $x<0$, $F(x)=1-e^{-x}$ for $x\ge 0$, the exponential distribution of rate one. Since this distribution function is continuous everywhere,

For the uniform distribution on $[0,1]$, the mean is $1/2$ and the variance is $1/12$. The central limit theorem for independent identically distributed variables with finite nonzero variance givesChoose $\boxed{\alpha =1/2,\beta =1/2}$. ThenThe standard normal distribution function is continuous and strictly increasing, so its value at the finite positive number $\sqrt{12}$ is strictly between $1/2$ and one, as required.