Defining $Y_{n}$ $=$ $nU^{n}$ where $U\sim$ UNIFORM$(0,1)$
How would one go about proving that $Y_{n}\to 0 $ a.s (as n$\to \infty $.) But does not converge to zero in $L^{p}$.
I can do the second part
$E(|Y_{n}-0|^{P})=n^{p} E(U^{np}) $ =
$$ n^{p} \int_{0}^{1} x^{np} dx= n^{p}\frac{1}{np+1}$$ $$\to \begin{cases} 1 & \quad \text{if } p \text{ is 1}\\ \infty & \quad \text{if } p \text{ >1} \end{cases} \text{neither are zero}$$
For the a.s part i have tried just using the definition but i get stuck with the limit, i can see how it does converge to $0$ but cant write a rigorous proof. Thanks