Nested Intervals
Definition: A sequence of intervals $I_n$ where $n \in \mathbb{N}$ is said to be nested if $I_1 \supseteq I_2 \supseteq I_3 \supseteq ... \supseteq I_n \supseteq I_{n+1} \supseteq ...$. |
For example, consider the interval $I_n = [0, \frac{1}{n} ]$ We note that $I_1 = [0, 1]$, $I_2 = [0, \frac{1}{2}]$, $I_3 = [0, \frac{1}{3} ]$, … As we can see $I_1 \supseteq I_2 \supseteq I_3 \supseteq ... \supseteq I_n \supseteq I_{n+1} \supseteq ...$ and so the sequence of intervals $I_n$ is nested. The diagram above illustrates this specific nesting of intervals.
Sometimes a nested interval will have a common point. In this specific example, the common point is $0$ since $0 ≤ 0 < \frac{1}{n}$ for all $n \in \mathbb{N}$. We denote the set theoretic intersection of all these intervals to be the set of common points in a nested set of intervals:
(1)Sometimes a set of nested intervals does not have a common point though. For example consider the set of intervals $I_n = (n, \infty)$. Clearly $I_1 \supseteq I_2 \supseteq I_3 \supseteq ... \supseteq I_n \supseteq I_{n+1} \supseteq ...$ since $I_1 = (1, \infty)$, $I_2 = (2, \infty)$, … However, there is no common point for these intervals.
Example 1
Determine the set $C$ of points to which the set of nested intervals $I_n = (1 - \frac{1}{n}, 2 + \frac{1}{n} )$ have in common.
We first note that $I_1 = [0, 3]$, $I_2 = [0.5, 2.5]$, $I_3 = [0.66..., 2.33...]$, … We conjecture that the set of points $C = \bigcap_{n=1}^{\infty} I_n = [1, 2]$ are contained within all the intervals. This can be informally deduced since $\lim_{n \to \infty} 1 - \frac{1}{n} = 1$ and $\lim_{n \to \infty} 2 + \frac{1}{n} = 2$.