Pointwise convergence

Pointwise convergence of a sequence of functions $(f_n)$ to a function $f$ on a set $X$ means: for every $x\in X$, the real sequence $(f_n(x))$ converges to $f(x)$, i.e.

This is a basic mode of convergence for sequences of functions , and it is weaker than uniform convergence (where one $N$ works simultaneously for all $x$). For each fixed $x$, pointwise convergence is just ordinary convergence of a sequence in the codomain.

Examples:

• On $[0,1]$, $f_n(x)=x^n$ converges pointwise to the function $f(x)=0$ for $0\le x<1$ and $f(1)=1$.
• On $\mathbb{R}$, $f_n(x)=\sin(nx)/n$ converges pointwise to $0$ (indeed $|f_n(x)|\le 1/n$ for all $x$).