Help with a Michael Spivak Calculus problem about sequence Consider a sequence of closed intervals $I_1 = [a_1, b_1], I_2 = [a_2, b_2]...$ Suppose $a_{n} \le a_{n+1}$ and $b_{n+1} \le b_n$ for all $n$. Prove that there is a point $x$, which is in every $I_n$.
I need some hints to begin. I could simply make the argument $x = a_n$ or $x = b_n$ since it is a closed-interval, but I want something more rigorous?
 A: Observe that $(a_n)$ an increasing sequence bounded above by $b_1$, and $(b_n)$ is a decreasing sequence bounded below by $a_1$. Hence, there are real numbers $a$ and $b$ such that $a = \lim_{n\to \infty} a_n$ and $b = \lim_{n\to \infty} b_n$. Since $a_n < b_n$ for all $n$, $a \le b$. Thus $a_n \le a \le b \le b_n$ for all $n$. Therefore, $a$ (and $b$) belongs to every $I_n$. 
A: The intervals are compact, and any finite intersection is non-empty. Define $J_k=I_k^c$. Then $J_k$ are open. Consider $I_1$. If no points of it belongs to every other $I_k$, then $J_2, J_3, \cdots$ is an open cover of $I_1$. Therefore, there is a finite subcover, let say $J_{k_1}, J_{k_2},\cdots, J_{k_n}$. But then $I_1\cap I_{k_1} \cap I_{k_2} \cdots \cap I_{k_n}=\phi$. A contradiction.
A: Let's get one thing clear: setting $x=a_n$ does not yet prove that there exists some value $x$ which is in every $I_n$. For example, if $$I_n=\left[-\frac{1}{n}, \frac 1n\right],$$ then $a_n=-\frac1n$ and therefore, no matter what the value of $n$ is, the statement 

$a_n$ is in $I_n$ for all values of $n$

is false, so your "proof" is not even an informal proof.
 Still, in this example, it is clear that $0$ is in all of the sets $I_n$, so this only shows your proof is wrong, the statement itself.

As for hints on how to begin, I advise you to consider the following points:


*

*The sequence $a_1, a_2, \dots$ is an increasing sequence

*The sequence $a_1, a_2, \dots$ is bounded above by $b_1$ (why? What would happen if there existed one such $a_m$ that $a_m > b_1$? Consider the relation between $b_1$ and $b_m$ in that case).

*The sequence $b_1, b_2, \dots$ is a decreasing sequence

*The sequence $b_1, b_2\dots$ is bounded below by $a_1$ (why)?


Now, what can you say about the two sequences? Do they converge? If yes, what can you say about their limits?
