Definition:Strictly Increasing/Sequence

Definition

Let $\left({S, \preceq}\right)$ be a totally ordered set.

Then a sequence $\left \langle {a_k} \right \rangle_{k \in A}$ of terms of $S$ is strictly increasing iff:

$\forall j, k \in A: j < k \implies a_j \prec a_k$

Real Sequences

The above definition for sequences is usually applied to real number sequences.

Let $\left \langle {x_n} \right \rangle$ be a sequence in $\R$.

Then $\left \langle {x_n} \right \rangle$ is strictly increasing if

$\forall n \in \N: x_n < x_{n+1}$

Also see

• Increasing Sequence
• Strictly Decreasing Sequence
• Strictly Monotone Sequence

Sources

• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 4.15$