Given $a_n = 0, b_n = \frac{1}{n} \forall n\geq1$
Does this imply the sequence of nested open intervals $((a_n, b_n))^{\infty}_{n = 1}$ has the condition satisfied that $\cap^{\infty}_{n=1} (a_n, b_n) = \emptyset$ ??
A user on a different thread claimed it did, my question is isn't the value $0$ part of the intersection? While $0$ is not explicitly in any sequence because the intervals are open, they go from (0,1), (0, 1/2), (0,1/3), ..... , (0, 1/99999999) and so forth, they each share that initial starting point. Hence their intersection cannot be $\emptyset$, can it?
Hint: Assume some $x>0$ is in the intersection. Can you find a contradiction to this assumption? That is, can you find one of the intervals in the intersection not containing $x$?
Responding to your edit: 0 is not in any of the sets under consideration, so it cannot be in the intersection.