1.3: Sequences
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 \(N=\{1,2,3, \ldots\}\).