Definition:Monotone (Order Theory)/Sequence/Real Sequence
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Definition
Let $\sequence {x_n}$ be a sequence in $\R$.
Then $\sequence {x_n}$ is monotone if and only if it is either increasing or decreasing.
Also known as
This can also be called a monotonic sequence.
Examples
Example: $\sequence {\paren {-1}^n}$
The first few terms of the real sequence:
- $S = \sequence {\paren {-1}^n}_{n \mathop \ge 1}$
are:
- $-1, +1, -1, +1, \dotsc$
$S$ is not monotone, either increasing or decreasing.
Also see
Sources
- 1953: Walter Rudin: Principles of Mathematical Analysis ... (previous) ... (next): $3.13$
- 1962: Bert Mendelson: Introduction to Topology ... (previous) ... (next): $\S 2.5$: Limits: Exercise $4$
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.15$: Sequences
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $\S 1.2$: Real Sequences: Definition $1.2.5$
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 4$: Convergent Sequences: $\S 4.15$: Monotone Sequences