Assume $Z_1, Z_2, Z_3,...$ are independent and identically distributed R.V.s s.t. $Z_n∈(-1,1)$. Prove the following:
1) $Z_n \to 0$ as $n \to \infty$. almost surely.
2) $Z_n \to 0$ as $n \to \infty$. in $L_1$.
I think i should start with the definition of almost sure convergence and fix an event w and observe the sequence $Z_1(w), Z_2(w), Z_3(w),...$, but what event should i fix on? and for L1, since its L1, we have to prove the expectation of $Z_n $converges to 0 but I dont think it is uniform distribution so how would i approach that?
Both are false. Obvious counter-example $Z_n=\frac 1 2$ for all $n$. Independent sequences rarely converge.