Throw dice, what does this mathematical expression mean in real life?

Question

Assuming we have a dice and the event that if we throw dice for the k-th time and get a 6 is given by $A_k$, is there an actual explanation what $A:= \cap_{i=1}^{\infty} \cup_{j=i}^{\infty} A_j$ is?

Answer 1

The inner part, $\cup_{j=i}^{\infty}A_j$, means "$6$ is thrown on the $i$-th or some later throw". So the full expression, $\cap_{i=1}^{\infty}\cup_{j=i}^{\infty}A_j$, means that "for each $i$, $6$ is thrown on the $i$-th or some later throw", or, more briefly, "infinitely many $6$s are thrown".