sequence

In standard mathematical convention, sequences are most often denoted using parentheses, e.g. \[\left( a_{n} \right)_{n\in\mathbb{N}}\], which denotes a sequence whose \[n\]th element is given by the variable \[a_{n}\]. For example, \[a_{1}\] is the first element of \[\left( a_{n} \right)_{n\in\mathbb{N}}\], \[a_{2}\] is the second element, \[a_{n}\] is the \[n\]th element, and so on. In most cases, a sequence is usually written as \[\left( a_{n} \right)\] as is often understood that the index \[n\] runs from 1 to \[\infty\].

The following notations are all equivalent: \[\left( a_{n} \right)^{\infty}_{n=1}=\left( a_{n} \right)_{n\in\mathbb{N}}=\left( a_{n} \right)\].

The variable \[n\] is known as the index, while the values that \[n\] can take (usually 1, 2, 3…) is called the index set.

There a special type of sequence known as a bi-infinite sequence, or \[\left( a_{n} \right)^{\infty}_{n=-\infty}\] and can also be written as \[\left( \dots,a_{-1},a_{0},a_{1},a_{2},\dots \right)\].

Do keep in mind that \[\left( a_{n} \right)\] and \[\left\{ a_{n} \right\}\] have slightly different meanings. The parentheses in \[\left( a_{n} \right)\] visually emphasize sequence and order, while curly braces, \[\left\{ a_{n} \right\}\] typically denote sets. In simpler words, \[\left( 1,2,3 \right)\ne \left( 2,3,1 \right)\] while \[\left\{ 1,2,3 \right\}=\left\{ 2,3,1 \right\}\].

When discussing sequences in analysis, sequences are generally considered as \[\left( x_{1},x_{2},x_{3},\dots \right)\], which is to say, infinite sequences of elements indexed by natural numbers.

The same also applies to subsequences, it is generally considered to be infinite unless specified otherwise.

A sequence can be defined using recursion if it's related to the previous elements in the sequence. A classic example for such a sequence would be the Fibonacci's sequence, where \[a_{n}=a_{n-1}+a_{n-2}\].

Similar to monotonic functions, a sequence is monotonically increasing if each term is greater than or equal to the one before it, which basically means the numbers in the sequence never decreases. For example, the sequence \[\left( a_{n} \right)\] is monotonically increasing if and only if \[a_{n+1}\ge a_{n}\] for all \[n\in\mathbb{N}\]. If the sequence fulfills the condition that the next element always increases, or \[a_{n+1}>a_{n}\], then we can classify it as a strictly monotonically increasing sequence.

A sequence is said to be monotonically decreasing if each term is smaller or equal to the term before it, or \[a_{n+1}\le a_{n}\], and strictly monotonically decreasing if \[a_{n+1}<a_{n}\].

If all the terms in a sequence of real numbers \[\left( a_{n} \right)\] are less than some real number \[M\] (not infinity), then the sequence is said to be bounded from above. In other words, if there exists \[M\] such that \[a_{n}\le M\] for all \[n\], then \[M\] is the upper bound of the sequence \[\left( a_{n} \right)\].

Likewise, if there exists some real number \[m\], such that \[a_{n}\ge m\] for all \[n\], then \[m\] is the lower bound of the sequence and thus the sequence is bounded from below.

If we can find \[M\] and \[m\] for a sequence, then the sequence is said to be bounded.

An example of such a sequence would be \[\left( n^{2} \right)_{n=1}^{10}\], when expanded would look like this, \[\left( 1,4,9,\dots,100 \right)\]. This sequence is both monotonically increasing (strictly, in fact), and bounded. It's lower bound, \[m\], would be 1, while the upper bound, \[M\], would be 100.

A subsequence of a sequence is a sequence formed from the given sequence by deleting some of the elements without disturbing the relative positions of the remaining elements.

For example, using back our sequence \[\left( n^{2} \right)_{n=1}^{10}\], a valid subsequence would be \[\left( 4,16,36,64,100 \right)\]. We've deleted all odd elements, without disturbing the relative positions of the remaining even elements.

Formally, a subsequence of the sequence \[\left( a_{n} \right)_{n\in\mathbb{N}}\] is any sequence of the form \[\left( a_{n_{k}} \right)_{k\in\mathbb{N}}\] where \[\left( n_{k} \right)_{k\in\mathbb{N}}\]. In simple words, a subsequence of \[\left( a_{n} \right)\] is a new sequence \[\left( a_{n_{k}} \right)\], where you pick the terms from \[\left( a_{n} \right)\] using an increasing sequence of indices \[\left( n_{k} \right)\] indexed by \[k\].