# 1.5. Sequences#

Definition 1.85 (Sequence)

Let $$A$$ be a set. Any function $$x : \Nat \to A$$, where $$\Nat = \{1,2,3,\dots\}$$ is the set of natural numbers, is called a sequence of $$A$$.

We say that $$x(n)$$ denoted by $$x_n$$ is the $$n^{\text{th}}$$ term in the sequence. We denote the sequence by $$\{ x_n \}$$.

We also write the sequence as:

$\{ x_1, x_2, \dots, \}.$

We can assign a symbol to a sequence as:

$X \triangleq \{ x_1, x_2, \dots, \}.$

Note that sequence may have repeated elements and the order of elements in a sequence is important. Tuples (like $$(a,b,c,d)$$) are also ordered but they are finite length. Sequences are ordered and of infinite (but countable) length.

Remark 1.19

We shall abuse the notation and use $$X$$ to denote both the sequence $$\{x_n \}$$ and the set of elements in the sequence. Thus, $$X$$ as a set is a subset of $$A$$. By $$x \in X$$, we shall mean that there exists a $$k \in \Nat$$ such that $$x$$ is the k-th element of the sequence $$X = \{ x_n \}$$.

Definition 1.86 (Eventually)

We say that a sequence $$\{ x_n \}$$ of a set $$A$$ satisfies a property $$(P)$$ eventually if there exists some natural number $$n_0$$ such that $$x_n$$ satisfies the property $$(P)$$ for all $$n > n_0$$.

Example 1.18

Consider the sequence $$\{ x_n = \frac{1000}{n} \}$$. For all $$n > 1000$$, it satisfies the property that $$x_n < 1$$.

Definition 1.87 (Subsequence)

A subsequence of a sequence $$\{ x_n \}$$ is a sequence $$\{ y_n \}$$ for which there exists a strictly increasing sequence $$\{ k_n \}$$ of natural numbers (i.e. $$1 \leq k_1 < k_2 < k_3 < \ldots)$$ such that $$y_n = x_{k_n}$$ holds for each $$n$$.