1 Sequences
1.1 Sequences
A sequence is an ordered indexed collection of elements or terms \(\{x_{n}\}_{n\in \mathcal{N}}\) where \(\mathcal{N}\) is some at most countably infinite index set. In Functional Analysis this is equivalent to a function whose domain is an interval of integers.
A sequence is said to be finite if \(\mathcal{N}\) is finite, otherwise the sequence is said to be infinite. A finite sequence of length \(n\) is known as an \(n\)-tuple. Sometimes we specify that a sequence is singly-infinite, that is, infinite in only one direction (e.g. \(0,1,2,3,4,\dots\)), or bi-infinite, infinite in both directions \((e.g. \dots,-2,-1,0,1,2,\dots)\).
In general we can always replace any index set \(\mathcal{N}\) with the set of natural numbers \(\mathbb{N}\).
1.1.1 Monotonicity
A sequence \(\{ x_{n} \}\) is said to be monotonically increasing if each term is greater than or equal to the one before it, that is \[ x_{n+1}\geq x_{n};~~\forall n. \] If we can drop the equality \(x_{n+1}>x_{n}\) for all \(n\) we say that the sequence is strictly monotonically increasing.
Similarly, a sequence \(\{ x_{n} \}\) is said to be monotonically decreasing if each term is less than or equal to the one before it, that is \[ x_{n+1}\leq x_{n};~~\forall n. \] If we can drop the equality \(x_{n+1}<x_{n}\) for all \(n\) we say that the sequence is strictly monotonically decreasing.
If a sequence is either increasing or decreasing it is known as a monotone sequence. This is a special can of the more general [[monotone-functions]].
Note: Sometimes we replace the terminology increasing with non-decreasing and decreasing with non-increasing to help avoid any possible confusion with strictly increasing and strictly decreasing.
A subsequence of \(\{ x_{n} \}_{n\in \mathbb{N}}\) is any sequence of the form \(\{ x_{n_{k}} \}_{k \in \mathbb{N}}\) where \(\{ n_{k} \}_{k \in \mathbb{N}}\) is a strictly increasing sequence of positive integers.
1.1.2 Boundedness
Consider a sequence \(\{ x_{n} \}\) on some ordered set such as the extended real numbers \(\overline{\mathbb{R}}:=\mathbb{R}\cup \{ \pm \infty \}\). The number \(u \in \mathbb{R}\) is said to be an upper bound of the sequence if \(x_{n}\leq u\) for all \(n \in \mathbb{N}\). We say that the sequence is bounded above. Similarly, the number \(l \in \overline{\mathbb{R}}\) is said to be lower bound of the sequence if \(x_{n}\geq l\) for all \(n \in \mathbb{N}\). We say that the sequence is bounded below. A sequence is bounded if it is both bounded above and below.