The nested interval theorem

By the Monotone Convergence Principle, the bounded monotone sequences $\{a_n\}$ and $\{b_n\}$ converge to limits $a$ and $b$ respectively. The length of the $n$-th interval is $L_n = b_n - a_n$.

Conditions: $\{a_n\}$ and $\{b_n\}$ are convergent sequences with limits $a$ and $b$.; The length of the intervals $b_n - a_n$ approaches 0 as $n \to \infty$.

The limits of the left and right endpoints converge to the same point because the length of the nested intervals approaches zero as $n$ goes to infinity. Since the distance between the endpoints vanishes, their respective limits must be equal, establishing a unique common point $\xi$.

Conditions: The intervals are nested closed intervals $[a_n, b_n]$.; The length of the intervals $b_n - a_n$ approaches zero as $n \to \infty$.; The sequences of endpoints converge to limits $a$ and $b$ respectively.

The sequence $\{b_n\}$ must be monotone decreasing (non-increasing) and bounded below. In the context of nested intervals, $b_{n+1} \le b_n$ follows from the containment $[a_{n+1}, b_{n+1}] \subseteq [a_n, b_n]$, ensuring monotonicity.

Conditions: Intervals are nested: $[a_{n+1}, b_{n+1}] \subseteq [a_n, b_n]$.

The uniqueness of the common point is proven by contradiction. Assuming there is another distinct point $\eta$ shared by all intervals leads to a logical conflict because the intervals shrink down to a single point $\xi$, making it impossible for $\eta$ to remain inside all intervals if $\eta \neq \xi$.

Conditions: The intervals are nested closed intervals $[a_n, b_n]$.; The length of the intervals approaches zero as $n \to \infty$.; There is a common point $\xi$ belonging to all intervals.

The Monotone Convergence Principle is applied to the sequences of left and right endpoints of the nested intervals. The sequence of left endpoints is monotonically increasing and bounded above, while the sequence of right endpoints is monotonically decreasing and bounded below.

Conditions: The intervals are nested closed intervals $[a_n, b_n]$.; The sequence of left endpoints $\{a_n\}$ is monotonically increasing and bounded above.; The sequence of right endpoints $\{b_n\}$ is monotonically decreasing and bounded below.

The inclusion condition $[a_{n+1}, b_{n+1}] \subseteq [a_n, b_n]$ implies two inequalities: $a_n \le a_{n+1}$ and $b_{n+1} \le b_n$. The inequality $a_n \le a_{n+1}$ means the sequence of left endpoints is non-decreasing (monotonic).

Conditions: A sequence of closed intervals satisfies $[a_{n+1}, b_{n+1}] \subseteq [a_n, b_n]$ for all $n$.