When applying the Monotone Convergence Principle to the right endpoints $\{b_n\}$, what specific properties must the sequence satisfy?

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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. Boundedness below is ensured because $b_n \ge a_1$ for all $n$ (since $a_1$ is the smallest possible left bound, and actually $b_n \ge a_n \ge a_1$ is not quite right, rather $b_n \ge a_n$ and $a_n$ is increasing, so $b_n$ is bounded below by any $a_k$, specifically $a_1$ works as a loose lower bound, or more precisely, the entire sequence is trapped between $a_1$ and $b_1$). Actually, simpler: $b_n \ge a_n \ge a_1$, so $b_n$ is bounded below by $a_1$.

When applying the Monotone Convergence Principle to the right endpoints $\{b_n\}$ , what specific properties must the sequence satisfy?

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