Let $z_1, z_2, z_3,$ ...be a sequence of independent random variables s.t. $P(z_i=1)=P(z_i=-1)=\frac 1 2$. Does this sequence converge almost surely?
My intuition is telling me that the sequence will approach the random variable Z? i.e. $z_n->Z$ as $n->\infty$ and Z is just another RV s.t. $P(Z=1)=P(Z=-1)=\frac 1 2$.
Here is my confusion, if a sequence converges almost surely, then $z_i$ should be different for each term s.t. it approaches Z in the end; but in this sequence $z-i$ is the same for every term. So I dont udnersntand.