7 January 2005
What is the Boundedness and Monotonicity
of a Sequence?
George E. Hrabovsky, President, MAST
Last time I wrote about the idea of a limit and the notion of sequences. I also introduced some standard sequences.
According to reference [1], if every element of a sequence is less than a particular value, then we can say that the value is the upper bound of the sequence. A sequence that has an upper bound is said to be bounded from above. If every element of a sequence is larger than a particular value then we say that the value is the lower bound of the sequence. A sequence that has an upper bound is said to be bounded from below. A sequence that is bounded from above and from below is said to be bounded.
It turns out that any sequence that has an upper bound has a least upper bound, that is a lowest value that is larger than any element of the sequence. We can see that this must be true, since the real numbers extend forever, and, if there is an upper bound, every real number above that value is also an upper bound. This value must be the next real number above the highest value of the sequence.
Similarly, there is also a greatest lower bound if there is a lower bound.
Beginning with an initial element of a sequence, if every subsequent element is smaller than the first, we say that the sequence is monotone decreasing. If every subsequent element is either the same or smaller than the preceding element, the sequence is monotone nonincreasing. If every subsequent element is larger, then the sequence is monotone increasing. If every subsequent element is equal or larger than its predecessor, the sequence is monotone nondecreasing. If any of these conditions is present, the sequence is called monotonic.
Next time we will explore what these concepts have to do with limits.
[1] Donald W. Hight, A Concept of Limits, Prentice-Hall, Inc. (1966, republished in 1977 by Dover Publications).
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(December 29, 2004)  |
