Monotone sequence of functions

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

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 .

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 $f(1)=1$.