Monotonicity of Real Sequences

Theorem
Let $\left\langle{a_n}\right\rangle: \mathbb D \to \R$ be a real sequence, where $\mathbb D$ is a subset of $\N$.

Let $\Bbb X$ be a real interval such that $\Bbb D \subseteq \Bbb X$.

Let $f: \Bbb X \to \R, x \mapsto f \left({x}\right)$ be a differentiable real function.

Suppose that for every $n \in \mathbb D$:


 * $f \left({n}\right) = a_n$

Then:


 * If $\forall x \in \Bbb X: D_x f \left({x}\right) \ge 0$, $\left\langle{a_n}\right\rangle$ is increasing


 * If $\forall x \in \Bbb X: D_x f \left({x}\right) > 0$, $\left\langle{a_n}\right\rangle$ is strictly increasing


 * If $\forall x \in \Bbb X: D_x f \left({x}\right) \le 0$, $\left\langle{a_n}\right\rangle$ is decreasing


 * If $\forall x \in \Bbb X: D_x f \left({x}\right) < 0$, $\left\langle{a_n}\right\rangle$ is strictly decreasing

where $D_x$ denotes differentiation w.r.t $x$.

Proof
Consider the case where $D_x f \left({x}\right) \ge 0$

Let $n \in \N$ be in the domain of $\left\langle{a_n}\right\rangle$.

From Derivative of Monotone Function, the sign of $D_x f$ is indicative of the monotonicity of $f$.

Because Differentiable Function is Continuous and Continuous Function is Riemann Integrable, $D_x f$ is integrable.

Hence:

Then:

By hypothesis:

Hence the result, by the definition of monotone.

The proofs of the other cases are similar.

Also see

 * Restriction of Monotone Function is Monotone