Is this a sufficient proof for the nested interval theorem? 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 $a_n \wedge b_n \in I_n$.
 A: HINT: From the hypothesis that the intervals are nested you know that
$$a_1\le a_2\le a_3\le\ldots~\ldots\le b_3\le b_2\le b_1\;.$$
Thus, you have two monotonic sequences, $\langle a_k:k\in\Bbb Z^+\rangle$ and $\langle b_k:k\in\Bbb Z^+\rangle$. Moreover, both are bounded: $a_1$ and $b_1$ are lower and upper bounds for both sequences. What fundamental fact about bounded monotonic sequences do you know?
A: The Nested Interval Theorem is sometimes known as the Nested Interval Property.
Why could it be either a 'theorem' or a 'property'?
Answer: Because it is equivalent to several other properties that come up when you are trying to show the real numbers are "complete."
If you are going to consider it a theorem (as opposed to a property) then you should identify which other properties you have at your disposal in order to prove it.
Edit: Knowing you have the Least Upper Bound Property, here is a proof (of my own) that implies the Principle of Nested Closed Intervals: (replace $\mathbb{F}$ with $\mathbb{R}$)

