Cauchy sequence

A sequence $(q_j{)}_{j\in {\mathbb{Z}}_{>0}}$ in the set of rational numbers $\mathbb{Q}$ is a Cauchy sequence if for each $ϵ\in {\mathbb{Q}}_{>0}$ there exists $N\in {\mathbb{Z}}_{>0}$ such that $|q_j-q_k|<ϵ$ for $j,k\le N$.

It converges to $q_0$ if for each $ϵ\in {\mathbb{Q}}_{>0}$, there exists $N\in {\mathbb{Z}}_{>0}$ such that $|q_j-q_0|<ϵ$ for $j\ge N$. It is bounded if there exists some $M\in {\mathbb{Q}}_{>0}$ such that $|q_j|<M$ for each $j\in {\mathbb{Z}}_{>0}$. A sequence has $q_0$ as its limit if it converges to $q_0$. Cauchy sequences are both bounded and convergent.

Okay. What does this all mean? The terms in the sequence get progressively closer to a specific value $q_0$ as we increase in the number of terms $j$. No matter our choice of $ϵ$, there’s a point in the sequence beyond which all terms are within that threshold of each other.

And why is this useful? Cauchy sequences guarantee the existence of a limit for the sequence. As long as we increase the number of terms, we’re guaranteed to land within the threshold $ϵ$ even if we don’t exactly hit 0. It does guarantee that terms will eventually cluster very close together, which is important in mathematics when thinking about convergence.

More definitions

The set of Cauchy sequences in $\mathbb{Q}$ is denoted $\text{CS}(\mathbb{Q})$.

We say that two sequences $q,r\in \text{CS}(\mathbb{Q})$ are equivalent if the sequence $(q_j-r_j{)}_{j\in {\mathbb{Z}}_{>0}}$ converges to zero. If this is the case, we write $(q_j{)}_{j\in {\mathbb{Z}}_{>0}}\sim (r_j{)}_{j\in {\mathbb{Z}}_{>0}}$.

Real sequences