Question

What does it mean for a sequence to be monotone?

Answer 1

It means that the sequence is always either increasing or decreasing, it the terms of the sequence are getting either bigger or smaller all the time, for all values bigger than or smaller than a certain value.

Explanation:

Here is the precise definitions :

• A sequence `(x_n) in RR or CC, ninNN` is called monotone increasing `iff EEkinNN`such that `x_(n+1)>=x_n AAn>=k`.
• A sequence `(x_n) in RR or CC, ninNN` is called monotone decreasing `iff EEkinNN`such that `x_(n+1)<=x_n AAn>=k`.

Note also that `(x_n)` is said to be bounded `iff EE MinNN`such that `(x_n)<=MAA ninNN`.

In addition, `(x_n)` converges to a limit `x in RR or CC iff AA epsilon >0 EE NinNN >0` such that `|x_n-x| < epsilon AA n > N`.

Furthermore, there is a theorem which states that every bounded, momotonic sequence is convergent.