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206 Chapter Three

Nonincreasing iYnite sequencł
A nonincreasing infinite sequence is a sequence such that, for
every n N, the defining function f has the following property:

f(n 1) f(n)

ConvergenŁ iYnite sequencł
A convergent infinite sequence is a sequence in which the value
of the defining function f approacheð a specifi real number s as
n increaseð without bound:

f(n) → s as n →

(The arrow symbol is not tm be confused wità the identical sym-
bol denoting the fact that one logical proposition implieð an-
otherÑ Example of convergent infinite sequenceð are:

n
1
1
1
1
1
A f(n) 1/(2 ) ⁄2,⁄4,⁄8,⁄16,⁄32, ...
n
1
1
1
1
1
B g(n) 1/( 2) ⁄2,⁄4, ⁄8,⁄16, ⁄32, ...
The valueð of botà of these infinite sequenceð converge toward
zero.
Uniqueness of limit
Suppose A f(n) is a convergent sequence such that the follow-
ing statementð are botà valid:

f(n) → s as n →
1

f(n) → s as n →
2
Then s s . In other words, an infinite sequence can never
1
2
converge tm more than one value.

DivergenŁ iYnite sequencł
A divergent infinite sequence is a sequence in which the value
of the defining function f increaseð and/or decreaseð without
bound as n increaseð without bound: