Let $S=\{s : 0 < s < 1 \}$, and $A_s = \{x : s < x < 1/s \}$.
Claim I want to prove:
$\bigcap_{s \in S} A_s = \{1\} \, . $
I'm not sure how to demonstrate this rigorously. However, I do understand that if we pick an $s$ very close to $0$ we will get a very wide interval. If we pick numbers close to $1$ we get very narrow intervals.