How does the nesting of intervals $[a_{n+1}, b_{n+1}] \subseteq [a_n, b_n]$ ensure that the sequence of left endpoints $\{a_n\}$ is monotonic and bounded?

Conditions

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

Reasoning, step by step

1. Identify the geometric meaning of set inclusion for intervals on the real line.
2. Deduce that the lower bound of the inner interval must be greater than or equal to the lower bound of the outer interval: $a_n \le a_{n+1}$.
3. Recognize this as the definition of a monotone increasing sequence.
4. Observe that because each subsequent interval is inside the previous ones, $a_n$ can never exceed the upper bound of the initial interval $b_1$.
5. Conclude that $\{a_n\}$ is monotone increasing and bounded above.

Example

If $[a_2, b_2] = [0.5, 0.8]$ is inside $[a_1, b_1] = [0, 1]$, then $a_2=0.5 \ge a_1=0$ (increasing) and $a_2=0.5 \le b_1=1$ (bounded above).

Common misconceptions

• Believing that nesting implies strict increase ($a_n < a_{n+1}$); weak inequality ($\le$) is sufficient for convergence proofs.
• Confusing the bounds; thinking $a_n$ is bounded below by $a_1$ is true but irrelevant for the Monotone Convergence Principle which requires an upper bound for increasing sequences.

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Explore next

• Why is the upper bound specifically $b_1$ and not some other value?
• Does the same logic apply to the right endpoints $\{b_n\}$?