Sequences

Sequence

A sequence of real numbers is a function $f:\mathbb{N}\to \mathbb{R}$, denoted $\{a_n{\}}_{n\in \mathbb{N}}$ with $a_n=f(n)$.

Increasing, Decreasing & Monotone

A sequence $\{a_n\}$ over $\mathbb{R}$ is increasing if $a_n\le a_{n+1}$, decreasing if $a_n\ge a_{n+1}$. $\{a_n\}$ is monotonic if it is either increasing or decreasing.

Alternating

A sequence $\{a_n\}$ over $\mathbb{R}$ is alternating if $a_n\cdot a_{n+1}<0$ for all $n\in \mathbb{N}$.

Monotone Convergence of Sequences

If a sequence $a_n$ is bounded and ultimately monotone, then it converges.