$X_1, X_2, X_3,\dots$ is a sequence of random signs, does this sequence converge almost surely?
I know the answer is no and I can use Cauchy in prob to prove it. However, I am very confused about the definition of convergence a.s.
Definition should be $P(w ∈ \Omega, lim X_n(w) \to X(w))=1$; now this is very trivial but obviously the sequence of random signs converges a.s. to a random sign using this definition: $P(w_1 ∈ \Omega, lim X_n(w_1) \to X(w_1))=1/2$, $P(w_2 \in \Omega, lim X_n(w_2) \to X(w_2))=1/2$ if $X(w_1)=1/2$, $X(w_2)=-1/2$. The question didn't specify what it converges to so I can specify it converges a.s. to X defined as another random sign. And I am using the definition of a.s. convergence here, not convergence in distribution.
Please tell me what the flaw in my logic is.