A sequence of real numbers is simply a function . For instance, the function defined on corresponds to the sequence .
Intuitively, a sequence converges if its terms approach a particular number.
Formally, a sequence of reals converges to if and only if for all positive reals , there exists a positive integer such that for all integers , we have . If converges to , is called the limit of and is written . The statement that converges to can be written as .
A classic example of convergence would be to show that as .
Proof: Let be arbitrary and choose . Then for we see that
which proves that , so as
Many significant sequences have their terms continually increasing, such as , or continually decreasing, such as . This motivates the following definitions:
A sequence of reals is said to be
- increasing if for all and strictly increasing if for all ,
- decreasing if for all and strictly decreasing if for all ,
- monotone if it is either decreasing or increasing.
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