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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 Monotone Convergence Theorem states that if a sequence $(a_n)_{n\in\mathbb{N}}$ is monotonically decreasing (i.e., $a_{n+1} \le a_n$ for all $n$) and bounded from below, then the sequence is convergent. This is a direct application of the existence of the infimum for bounded sets.

Conditions: The sequence $(a_n)$ is monotonically decreasing.; The sequence $(a_n)$ is bounded from below.

The Monotone Convergence Theorem states that if a sequence $(a_n)_{n\in\mathbb{N}}$ is monotonically decreasing (i.e., $a_{n+1} \le a_n$ for all $n$) and bounded from below, then the sequence is convergent. This is a direct application of the existence of the infimum for bounded sets.

Conditions: The sequence $(a_n)$ is monotonically decreasing.; The sequence $(a_n)$ is bounded from below.

The distance between any two terms $b_n$ and $b_m$ (with $m > n$) is bounded by the length of the interval $[a_n, b_n]$. Because the bisection method halves the interval length at each step, this distance is bounded by $\left(\frac{1}{2}\right)^{n-1}|b_1 - a_1|$.

Conditions: $m > n \in \mathbb{N}$.; The sequences $(a_n)$ and $(b_n)$ are constructed via the bisection method.; $b_n$ is an upper bound and $a_n$ is a lower bound for the set $M$.

The distance between any two terms $b_n$ and $b_m$ (with $m > n$) is bounded by the length of the interval $[a_n, b_n]$. Because the bisection method halves the interval length at each step, this distance is bounded by $\left(\frac{1}{2}\right)^{n-1}|b_1 - a_1|$.

Conditions: $m > n \in \mathbb{N}$.; The sequences $(a_n)$ and $(b_n)$ are constructed via the bisection method.; $b_n$ is an upper bound and $a_n$ is a lower bound for the set $M$.