Monotone sequence of functions - Imported Knowlpedia Content

A sequence of real-valued functions $(f_n)$ on a set $X$ is a monotone increasing sequence of functions if

f_n(x)\le f_{n+1}(x)\quad \text{for all } x\in X \text{ and all } n,

and it is monotone decreasing if $f_n(x)\ge f_{n+1}(x)$ for all $x$ and $n$ . Equivalently, for each fixed $x\in X$ , the numerical sequence $(f_n(x))$ is a monotone sequence.

Remarks

Monotone sequences of functions are typically studied together with pointwise convergence, and on compact domains they feature in Dini's theorem (which upgrades certain monotone pointwise limits to uniform convergence).

Examples

On any $X\subseteq\mathbb{R}$ , the functions $f_n(x)=x-\frac1n$ form a monotone increasing sequence (pointwise) with limit $f(x)=x$ .
On $[0,1]$ , the functions $f_n(x)=x^n$ form a monotone decreasing sequence (pointwise) with pointwise limit $f(x)=0$ for $x<1$ and $f(1)=1$ .