Monotony of Real Sequences

Monotony Criteria

Theorem: Difference Criteria for Sequence Monotony

A real sequence $(a_n{)}_{n\in \mathbb{N}}$ is

• increasing if $a_{n+1}-a_n\ge 0$ for all $n\in \mathbb{N}$;

• strictly increasing if $a_{n+1}-a_n>0$ for all $n\in \mathbb{N}$;

• decreasing if $a_{n+1}-a_n\le 0$ for all $n\in \mathbb{N}$;

• strictly decreasing if $a_{n+1}-a_n<0$ for all $n\in \mathbb{N}$

PROOF

TODO

Theorem: Quotient Criteria for Sequence Monotony

A real sequence $(a_n{)}_{n\in \mathbb{N}}$ with $a_n>0$ for all $n\in \mathbb{N}$ is:

• increasing if $\frac{a_{n+1}}{a_n}\ge 1$ for all $n\in \mathbb{N}$;

• strictly increasing if $\frac{a_{n+1}}{a_n}>1$ for all $n\in \mathbb{N}$;

• decreasing if $\frac{a_{n+1}}{a_n}\le 1$ for all $n\in \mathbb{N}$;

• strictly decreasing if $\frac{a_{n+1}}{a_n}<1$ for all $n\in \mathbb{N}$

PROOF

TODO

Convergence of Monotonous Sequences

Theorem: Convergence of Monotonic Sequences

An increasing real sequence is convergent if and only if it is bounded above.

A decreasing real sequence is convergent if and only if it is bounded below.

PROOF

TODO