Monotonicity of Real Sequences

Theorem
Let $\mathbb D$ be a subset of $\N$.

Let $\sequence {a_n}: \mathbb D \to \R$ be a real sequence.

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

Let $f: \mathbb X \to \R, x \mapsto \map f x$ be a differentiable real function.

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


 * $\map f n = a_n$

Then:


 * If $\forall x \in \mathbb X: D_x \map f x \ge 0$, $\sequence {a_n}$ is increasing


 * If $\forall x \in \mathbb X: D_x \map f x > 0$, $\sequence {a_n}$ is strictly increasing


 * If $\forall x \in \mathbb X: D_x \map f x \le 0$, $\sequence {a_n}$ is decreasing


 * If $\forall x \in \mathbb X: D_x \map f x < 0$, $\sequence {a_n}$ is strictly decreasing

where $D_x$ denotes differentiation $x$.

Proof
Consider the case where $D_x \map f x \ge 0$

Let $n \in \N$ be in the domain of $\sequence {a_n}$.

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 Real Function is Darboux Integrable, $D_x f$ is integrable.

Hence:

Then:

Hence the result, by the definition of monotone.

The proofs of the other cases are similar.

Also see

 * Restriction of Monotone Function is Monotone