Let be $\{U_i\}_{i\geq 1}$ a family of random variables independent and with the same distribution, $U[0,1]$ (uniform on [0,1]). Define $Y_n=\max_{1\leq i \leq n}\frac{U_i}{i}$ and show that $Y_n$ converges to a known distribution and identify which one it is.
His CDF is $n! t^n$ and i did it like this. Since their are independet $$F_{Y_n}(t)=\mathbb{P}(Y_n \leq t)=\mathbb{P}(U_1 \leq t\cap...\cap U_n\leq nt)=\prod_{i=1}^n\mathbb{P}(U_i\leq it)=\prod_{i=1}^nit=t^n n!$$ Although i dont know if its all right, since I did not take into account the support of all $U_o$ and the support of $Y_n$. Then if $t<1$, clearly $F_{Y_n}(t) \longrightarrow 0$ as n tends to infinity.
Which is obviously wrong, I think that i mess up on the product, but i dont know how. Any help? I tried an approach with moment-generating function arriving to the same conclusion
