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 \(n\).