# 1.3: Sequences

- Page ID
- 19023

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By an *infinite sequence* (briefly *sequence*) we mean a mapping (call it \(u )\) whose domain is \(N\) (all natural numbers \(1,2,3, \dots ) ; D_{u}\) may also contain \(0 .\)

A *finite sequence* is a map \(u\) in which \(D_{u}\) consists of all positive (or non-negative) integers *less than a fixed integer *\(p .\) The *range* \(D_{u}^{\prime}\) of any sequence \(u\) may be an arbitrary set \(B ;\) we then call \(u\) a sequence of elements of \(B,\) or in \(B .\) For example,

\[u=\left( \begin{array}{lllllll}{1} & {2} & {3} & {4} & {\ldots} & {n} & {\ldots} \\ {2} & {4} & {6} & {8} & {\ldots} & {2 n} & {\dots}\end{array}\right)\]

is a sequence with

\[D_{u}=N=\{1,2,3, \ldots\}\]

and with function values

\[u(1)=2, u(2)=4, u(n)=2 n, \quad n=1,2,3, \ldots\]

Instead of \(u(n)\) we usually write \(u_{n}\) ("index notation"), and call \(u_{n}\) the \(n^{th}\) *term* of the sequence. If \(n\) is treated as a *variable*, \(u_{n}\) is called the *general term* of the sequence, and \(\left\{u_{n}\right\}\) is used to denote the entire (infinite) sequence, as well as its range \(D_{u}^{\prime}\) (whichever is meant, will be clear from the context). The formula \(\left\{u_{n}\right\} \subseteq B\) means that \(D_{u}^{\prime} \subseteq B,\) i.e., that \(u\) is a sequence in \(B\). To

determine a sequence, it suffices to define its general term \(u_{n}\) by some formula or rule. **In** \((1)\) **above**, \(u_{n}=2 n\).

Often we omit the mention of \(D_{u}=N\) (since it is *known*) and give only the range \(D_{u}^{\prime} .\) Thus instead of \((1),\) we briefly write

\[2,4,6, \ldots, 2 n, \ldots\]

or, more generally,

\[u_{1}, u_{2}, \dots, u_{n}, \dots\]

Yet it should be remembered that \(u\) is a set of *pairs* (a map).

If all \(u_{n}\) are *distinct* (different from each other), \(u\) is a *one-to-one *map. However, this need not be the case. It may even occur that all \(u_{n}\) are equal (then \(u\) is said to be constant); e.g., \(u_{n}=1\) yields the sequence \(1,1,1, \ldots, 1, \ldots,\) i.e.

\[u=\left( \begin{array}{cccccc}{1} & {2} & {3} & {\ldots} & {n} & {\ldots} \\ {1} & {1} & {1} & {\ldots} & {1} & {\ldots}\end{array}\right)\]

Note that here\(u\) is an *infinite* sequence (since \(D_{u}=N\)), even though its range \(D_{u}^{\prime}\) has only one element, \(D_{u}^{\prime}=\{1\} .\) (In *sets*, repeated terms count as *one* element; but the *sequence* \(u\) consists of infinitely many distinct *pairs \((n, 1) .\) ) If all \(u_{n}\) are real numbers, we call \(u\) a real sequence. For such sequences, we have the following definitions.*

Definition 1

A real sequence \(\left\{u_{n}\right\}\) is said to be *monotone* (or *monotonic*) iff it is either *nondecreasing*, i.e.

\[(\forall n) \quad u_{n} \leq u_{n+1}\]

or *nonincreasing*, i.e.,

\[(\forall n) \quad u_{n} \geq u_{n+1}\]

Notation: \(\left\{u_{n}\right\} \uparrow\) and \(\left\{u_{n}\right\} \downarrow,\) respectively. If instead we have the *strict* inequalities \(u_{n}<u_{n+1}\) (respectively, \(u_{n}>u_{n+1} ),\) we call \(\left\{u_{n}\right\}\) *strictly* *monotone* (increasing or decreasing).

A similar definition applies to sequences of *sets.*

Definition 2

A sequence of sets \(A_{1}, A_{2}, \ldots, A_{n}, \ldots\) is said to be *monotone* iff it is either *expanding*, i.e.,

\[(\forall n) \quad A_{n} \subseteq A_{n+1}\]

or *contracting*, i.e.,

\[(\forall n) \quad A_{n} \supseteq A_{n+1}\]

Notation: \(\left\{A_{n}\right\} \uparrow\) and \(\left\{A_{n}\right\} \downarrow,\) respectively. For example, any sequence of concentric solid spheres (treated as *sets of points*), with increasing radii, is expanding; if the radii decrease, we obtain a contracting sequence.

Definition 3

Let \(\left\{u_{n}\right\}\) be any sequence, and let

\[n_{1}<n_{2}<\cdots<n_{k}<\cdots\]

be a *strictly increasing* sequence of natural numbers. Select from \(\left\{u_{n}\right\}\) those terms whose subscripts are \(n_{1}, n_{2}, \ldots, n_{k}, \ldots\) Then the sequence \(\left\{u_{n_{k}}\right\}\) so selected (with \(k\) th term equal to \(u_{n_{k}} ),\) is called the *subsequence* of \(\left\{u_{n}\right\},\) determined by the subscripts \(n_{k}, k=1,2,3, \ldots\).

Thus (roughly) a subsequence is any sequence obtained from \(\left\{u_{n}\right\}\) by dropping some terms, *without changing the order of the remaining terms* (this is ensured by the inequalities \(n_{1}<n_{2}<\cdots<n_{k}<\cdots\) where the \(n_{k}\) are the subscripts of the remaining terms). For example, let us select from (1) the subsequence of terms whose subscripts are *primes* (including 1). Then the subsequence is

\[2,4,6,10,14,22, \dots\]

i.e.,

\[u_{1}, u_{2}, u_{3}, u_{5}, u_{7}, u_{11}, \dots\]

All these definitions apply to finite sequences accordingly. Observe that every sequence arises by "numbering" the elements of its range (the terms): \(u_{1}\) is the *first* term, \(u_{2}\) is the *second* term, and so on. By so numbering, we put the terms in a certain *order*, determined by their subscripts \(1,2,3, \ldots\) (like the numbering of buildings in a street, of books in a library, etc. \() .\) The question now arises: Given a set \(A,\) is it always possible to "number" its elements *by integers*? As we shall see in \(\$ 4,\) this is not always the case. This leads us to the following definition.

Definition 4

A set \(A\) is said to be *countable* iff \(A\) is contained in the range of some sequence (briefly, the *elements* of \(A\) *can be put in a sequence*).

If, in particular, this sequence can be chosen finite, we call \(A\) a *finite* set. (The empty set is finite.)

Sets that are not finite are said to be *infinite*.

Sets that are not countable are said to be *uncountable*.

Note that all finite sets are countable. The simplest example of an infinite countable set is \(N=\{1,2,3, \ldots\}\).