$X_k \sim U(0,L)$ i.i.d; $H_0: L=L_0 ~ H_1: L \ne L_0$; $L,a \in(0,1)$ is a test with the following power function:
$P_n(L_0)= P( max_{k=1,..,n}X_k>L_0~\cup~ max_{k=1,..n}X_k<L_0*a^{1/n})$. How do I show that this test is consistent? So far I only managed to use the properties of the $X_k$ and the fact that both sets are disjoint to arrive at $P_n(L_0)=1+(\frac{L_0}{L})^n(a-1)$. I'm not really sure how to show that the second term converges to 0, assuming it is possible at all and I don't need another approach entirely.