A sequence \((a_n)\) is a function \(a:\mathbb{N} \rightarrow \mathbb{R}\) that maps natural numbers \(n \in \mathbb{N}\) to real numbers \(a_n \in \mathbb{R}.\)

A sequence \((a_n)\) is bounded by real numbers \(A\) and \(B\) if and only if \(A \leq a_n \leq B \, \, \forall n.\) Equivalently, \((a_n)\) is bounded if and only if \(\exists M \geq 0 : \lvert a_n\rvert \leq M \, \, \forall n.\)

A sequence \((a_n)\) is monotonically increasing if and only if \(a_{n+1} \geq a_n \, \, \forall n\) and monotonically decreasing if and only if \(a_{n+1} \leq a_n \, \, \forall n.\)

A sequence \((a_n)\) converges to a real number \(L\) if and only if for every \(\varepsilon >0\) there exists \(N \in \mathbb{N}\) such that \(a_n\) is \(\varepsilon\)-close to \(L\) for all \(n \geq N.\)

\[\forall \varepsilon > 0 \, \exists N \in \mathbb{N} : \lvert a_n-L \rvert \leq \varepsilon \, \forall n \geq N\]

Every convergent sequence is bounded and every bounded monotonic sequence is convergent. A sequence is divergent if it is not convergent.

A sequence \((a_n)\) is a Cauchy sequence if for every \(\varepsilon > 0\) there exists \(N \in \mathbb{N}\) such that for all natural numbers \(m, n \geq N,\) \(a_m\) and \(a_n\) are \(\varepsilon\)-close.

\[\forall \varepsilon > 0 \, \exists N \in \mathbb{N} : \lvert a_n-a_m \rvert \leq \varepsilon \, \forall m,n \geq N\]

A Cauchy sequence is a sequence whose terms become arbitrarily close to one another. A sequence of real numbers is convergent if and only if it is a Cauchy sequence. This holds because the real numbers are complete; it fails in the rational numbers, where the decimal truncations \(1, 1.4, 1.41, 1.414, \ldots\) of \(\sqrt{2}\) form a Cauchy sequence with no rational limit.