Definition:Strictly Increasing/Sequence
Definition
Let $\struct {S, \preceq}$ be a totally ordered set.
Then a sequence $\sequence {a_k}_{k \mathop \in A}$ of terms of $S$ is strictly increasing if and only if:
- $\forall j, k \in A: j < k \implies a_j \prec a_k$
Real Sequence
The above definition for sequences is usually applied to real number sequences:
Let $\sequence {x_n}$ be a sequence in $\R$.
Then $\sequence {x_n}$ is strictly increasing if and only if:
- $\forall n \in \N: x_n < x_{n + 1}$
Also known as
A strictly increasing sequence is also referred to as ascending or strictly ascending.
Some sources refer to a strictly increasing sequence as an increasing sequence, and refer to an increasing sequence which is not strictly increasing as a monotonic increasing sequence to distinguish it from a strictly increasing sequence.
That is, such that monotonic is being used to mean an increasing sequence in which consecutive terms may be equal.
$\mathsf{Pr} \infty \mathsf{fWiki}$ does not endorse this viewpoint.
Also see
- Definition:Increasing Sequence
- Definition:Strictly Decreasing Sequence
- Definition:Strictly Monotone Sequence
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 18$: Induced $N$-ary Operations
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): increasing sequence
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): increasing sequence
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): increasing sequence