1.7. Sequences

Definition 1.93 (Sequence)

Let $A$ be a set. Any function $x:\mathbb{N}\to A$, where $\mathbb{N}=\{1,2,3,\dots \}$ 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 $k\in \mathbb{N}$ such that $x$ is the k-th element of the sequence $X=\{x_n\}$.

Definition 1.94 (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 1.95 (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. $1\le k_1<k_2<k_3<\dots )$ such that $y_n=x_{k_n}$ holds for each $n$.