Big Idea of Functions applied to sequence

Invoking the Big Idea of Functions

From the topic of Functions, recall that a function \(\mathrm{f}\) is a rule that assigns to each given \(x\) in the domain \(D_f\) a unique element \(\mathrm{f}(x)\) in its range \(R_\mathrm{f}\), which we denote by

\[\mathrm{f} : x \mapsto \mathrm{f}(x), \quad x \in D_\mathrm{f}.\]

A sequence \(\{u_n\}\) is actually a function \(u\) that assigns to each index \(n \in \mathbb{N}\) its \(n^{\text{th}}\) term \(u_n\) in \(\mathbb{R}\), i.e.,

\[u : n \mapsto u_n, \quad n \in \mathbb{N}.\]