What does the completeness axiom state about Cauchy sequences and convergent sequences in real numbers?

The standard convergence definition requires knowing the limit $a$ in advance because it measures the distance $|a-a_n|$ to a specific number. The Cauchy condition avoids this limitation by measuring the pairwise distance $|a_n-a_m|$ between sequence terms, allowing one to verify convergence-like behavior without needing to know the limit beforehand.

Conditions: The sequence is being tested for convergence-like properties.; The candidate limit $a$ is unknown or not provided in advance.

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 $\mathbb{R}$.

A sequence $(a_n)_{n\in\mathbb{N}}$ is a Cauchy sequence if for every positive $\varepsilon$, there exists a natural number $N$ such that for all indices $n,m\ge N$, the distance between the corresponding terms satisfies $|a_n-a_m|<\varepsilon$.

Conditions: The sequence consists of real numbers.; The condition must hold for every $\varepsilon>0$.; The threshold $N$ may depend on $\varepsilon$.; Both indices $n$ and $m$ must be greater than or equal to $N$.

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 $M$.