> [!Definition] Definiton 1 (Sequence of real numbers) > Let $\mathbb{R}$ denote the [[Real Numbers|set of real numbers]]. Let $a\in\mathbb{Z}$ and $A=\{ m \in \mathbb{Z}: m \geq a \}$. A [[Function|function]] $s:A\to \mathbb{R}$ is called a sequence of real numbers. > [!NOTE] Definition 2 (Sequence of real numbers) > A sequence of real numbers is a [[Sequence|sequence]] of [[Real Numbers|the set of real numbers]]. **Notation**: Given a sequence of real numbers $s:A \to \mathbb{R}$, we write $s_{n}$ instead of $s(n)$, where $n\in A$. Moreover we denote the sequence by $(s_{n})$ or $(s_{n})_{n=a}^{\infty}$ instead of $s_{a},s_{a+1},s_{a+2},\dots$ # Properties We may define a sequence by a [[Recurrence Relation|recurrence relation]]. A sequence is [[Convergent Real Sequence|convergent]] if it has a limiting value. Otherwise it may simply not converge (e.g. $a_{n}=(-1)^{n}$) or [[Properly Divergent Sequence|tend to infinity]]. A [[Monotonic Sequence of Real Numbers]] is one that is increasing or decreasing. A [[Bounded Sequence]] is one that is bounded. A [[Real Subsequence]] of a sequence is a sequence that contains some terms of the original in the same order. Every infinite sequence has a [[Monotonic Subsequence Theorem|monotonic subsequence]]. # Applications The infinite sum of the terms of a sequence is a [[Series of Real Sequence|series]].