The Nested Interval Property

In this post, I want to talk about the definition of completeness, and then use the completeness of the real numbers to prove the Nested Interval Property. Before we define completeness, we need the definition of a Cauchy sequence.

Definition: A sequence $\{x_1,x_2,\ldots,x_k,\ldots\}$ of real numbers is called Cauchy if for all $\epsilon>0$ , there exists an $N\in\mathbb{N}$ such that for all natural numbers $m,n>N$ , $|x_m-x_n|<\epsilon$ .

This definition says that a sequence is Cauchy if, for any positive tolerance (no matter how small), you can go far enough out in the sequence (past $N$ ) where any two elements past that certain point will be closer together than the tolerance you chose.

A good intuition for this definition is that, in a Cauchy sequence, all of the terms of the sequence get arbitrarily close together eventually (after some a given point). The point after which this happens depends on the tolerance you set (how close together you want all of the elements to be).

Now, let’s define what it means for a space to be complete.

Definition: A metric space $\mathbb{M}$ (for example, $(\mathbb{R},|\cdot|)$ ) is complete if every Cauchy sequence with elements in $\mathbb{M}$ converges to an element in $\mathbb{M}$ .

What this definition is really saying is that a space is complete if, whenever all of the terms are getting arbitrarily close to each other, they are also getting arbitrarily close to an element of the space that they’re in.

Now let’s talk about the Nested Interval Property. This is a statement about the real numbers that says…

Let $\{I_1,I_2,\ldots\}$ is a sequence of nested closed intervals, which look something like this:

Adapted from https://en.wikipedia.org/wiki/Nested_intervals

Then the infinite intersection $\bigcap_{n=1}^{\infty} [A_n,B_n]$ is non-empty. This property says that no matter how close you zoom in on the real number line, there will always be real numbers where you are looking.

Now, let’s use the completeness of the real numbers to prove the Nested Interval Property.

When we build the sequence of nested intervals $\{[A_1,B_1],[A_2,B_2],\ldots\}$ , we’re simultaneously constructing two sequences of real numbers: One increasing sequence $\{A_1,A_2,\ldots,\}$ , and one decreasing sequence $\{B_1,B_2,\ldots\}$ .

These sequences will definitely be Cauchy because the upper limit of the increasing lower bounds can never exceed the lower limit of the decreasing upper bounds. (i.e., $sup\{A_n\}\leq inf\{B_n\}$ ).

Since every Cauchy sequence converges, we know that both of these sequences will converge. And this guarantees that the infinite intersection $\cap_{n=1}^{\infty} I_n$ of the nested sets will be nonempty!