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The Monotone Convergence Theorem states that every bounded, monotone sequence of real numbers converges. Therefore, to prove convergence using this theorem, one must establish both boundedness and monotonicity as the necessary hypotheses.

Conditions: The sequence consists of real numbers.; The goal is to prove convergence using the Monotone Convergence Theorem.

The Monotone Convergence Theorem states that every bounded, monotone sequence of real numbers converges. Therefore, to prove convergence using this theorem, one must establish both boundedness and monotonicity as the necessary hypotheses.

Conditions: The sequence consists of real numbers.; The goal is to prove convergence using the Monotone Convergence Theorem.

The bisection method constructs a sequence of upper bounds $(b_n)$ that is proven to be a Cauchy sequence. However, being Cauchy only means the terms get close to each other; it does not guarantee they converge to a specific real number.

Conditions: The sequence $(b_n)$ has been proven to be a Cauchy sequence.; The context is the field of real numbers $\mathbb{R}$.

The bisection method constructs a sequence of upper bounds $(b_n)$ that is proven to be a Cauchy sequence. However, being Cauchy only means the terms get close to each other; it does not guarantee they converge to a specific real number.

Conditions: The sequence $(b_n)$ has been proven to be a Cauchy sequence.; The context is the field of real numbers $\mathbb{R}$.

The inequality $-\frac{2}{3}a_n+\frac{1}{3}\le0$ is algebraically equivalent to $\frac{1}{3}\le\frac{2}{3}a_n$, which simplifies to $\frac{1}{2}\le a_n$. Since the lower bound $a_n\ge\frac{1}{2}$ was already proven by induction, the difference $a_{n+1}-a_n$ is guaranteed to be non-positive.

Conditions: The expression for the difference is $a_{n+1}-a_n = -\frac{2}{3}a_n + \frac{1}{3}$.; The lower bound $a_n \ge \frac{1}{2}$ has been established for all $n \ge 1$.

The inequality $-\frac{2}{3}a_n+\frac{1}{3}\le0$ is algebraically equivalent to $\frac{1}{3}\le\frac{2}{3}a_n$, which simplifies to $\frac{1}{2}\le a_n$. Since the lower bound $a_n\ge\frac{1}{2}$ was already proven by induction, the difference $a_{n+1}-a_n$ is guaranteed to be non-positive.

Conditions: The expression for the difference is $a_{n+1}-a_n = -\frac{2}{3}a_n + \frac{1}{3}$.; The lower bound $a_n \ge \frac{1}{2}$ has been established for all $n \ge 1$.