Definition:Increasing/Sequence/Real Sequence
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Definition
Let $\sequence {x_n}$ be a sequence in $\R$.
Then $\sequence {x_n}$ is increasing if and only if:
- $\forall n \in \N: x_n \le x_{n + 1}$
Also known as
Some sources refer to such a sequence as (monotone) non-decreasing.
Examples
Example: $\sequence 1$
The first few terms of the real sequence:
- $S = \sequence 1_{n \mathop \ge 1}$
are:
- $1, 1, 1, 1, \dotsc$
$S$ is both increasing and decreasing.
Also see
- Definition:Strictly Increasing Real Sequence
- Definition:Decreasing Real Sequence
- Definition:Monotone Real Sequence
Sources
- 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: Definition $15.3$
- 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