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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 bisection method starts with an element $a_1$ in the set $M$ and an upper bound $b_1$. It iteratively calculates the midpoint $c_n = \frac{1}{2}(a_n + b_n)$.

Conditions: $M$ is a non-empty subset of $\mathbb{R}$.; $M$ is bounded from above.; $a_1 \in M$ and $b_1$ is an upper bound for $M$.

The bisection method starts with an element $a_1$ in the set $M$ and an upper bound $b_1$. It iteratively calculates the midpoint $c_n = \frac{1}{2}(a_n + b_n)$.

Conditions: $M$ is a non-empty subset of $\mathbb{R}$.; $M$ is bounded from above.; $a_1 \in M$ and $b_1$ is an upper bound for $M$.