I need some help. Sorry for the poor use of LaTEX...
a) $U_{n} = \lfloor{nU}\rfloor/n$ prove that $\lim_{n \to +\infty} {U_{n}} = U$ where $U \sim unif[0,1]$ and thus that $lim_{n \to +\infty} EU_{n}^{p} = EU^{p}$, $p > 0$
For this one I've shown that U - $U_{n} \leq $ 1/n and as n $\rightarrow + \infty$ and thus $EU_{n}^{p} \rightarrow EU^{p}$ as for x real $x^{p}$ is a continuous function and $lim_{n \to +\infty} Eg(U_{n}) = Eg(U)$ for g continuous.
b) Compute $P(U_{n+1} > U_{n})$ and $P(U_{n+1} = U_{n})$ for each $n \in \mathbb{N}$ to show that the convergence is not monotone.
I don't know where to start part b)