1.5. Sequences¶

Definition 1.85 (Sequence)

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

We say that $x (n)$ denoted by $x_{n}$ is the $n^{th}$ 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 ≜ {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 $k \in N$ 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. $1 \leq k_{1} < k_{2} < k_{3} < \dots)$ such that $y_{n} = x_{k_{n}}$ holds for each $n$ .

Topics in Signal Processing

Sequences

1.5. Sequences¶