A *sequence of real numbers* is a function whose domain is the set of natural numbers 1, 2,…, *k*,… and whose range is contained in . Thus, a sequence of real numbers can be viewed as a set of numbers {*x*_{1}, *x*_{2},…, *x*_{k},…}, which is often also denoted as {*x*_{k}} (or sometimes as , to indicate explicitly the range of values that *k* can take).

A sequence {*x*_{k}} is *increasing* if *x*_{1} < *x*_{2} < · · · < *x*_{k} · · ·; that is, *x*_{k} < *x*_{k+1} for all *k*. If *x*_{k} ≤ *x*_{k+1}, then we say that the sequence is *nondecreasing*. Similarly, we can define *decreasing* and *nonincreasing sequences*. Nonincreasing or nondecreasing sequences are called *monotone sequences*.

A number *x** is called the *limit* of the sequence {*x*_{k}} if for any positive ε there is a number *K* (which may depend on ε) such that for all *k* > *K*, |*x*_{k} − *x**| < ε; that is, *x*_{k} lies between *x** − ε and *x** + ε for all *k* > *K*. In this case we write

or

A sequence that has a limit is called a *convergent sequence*.

The notion of a sequence ...

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