# Sequences

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 \(A< \lvert a_n\lvert<B \, \, \forall n.\)

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 \(\exists N \in \mathbb{N}\) such that \((a_n)\) is \(\varepsilon\)-close to \(L\) for all \(\varepsilon >0.\) 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 there exists a positive integer \(N \in \mathbb{N}\) such that for all natural numbers \(m, n \geq N,\) \(a_m\) and \(a_n\) are \(\varepsilon\)-close.

A Cauchy sequence is a sequence whose terms become arbitrarily close to one another. A sequence is **convergent** if and only if it is a Cauchy sequence.