Q: Consider a sequence of closed intervals $I_1 = [a_1, b_1], I_2 = [a_2, b_2], \dots$. Suppose that $\forall n \in \mathbb{N} \left(a_n \leq a_{n + 1} \wedge b_{n + 1} \leq b_n \right)$. Prove that there is a point $x$ in every $I_n$.
Proof: This is equivalent to proving that $\forall n$ we have some $x$ such that $x \in I_n$. Trivially we have $a_n \wedge b_n \in I_n$.