Metric space/Mapping sequence/Pointwise convergent/Definition

Pointwise convergent

Let ${}T$ denote a set, ${}M$ a metric space, and

$f_{n}\colon T\longrightarrow M$

(${}n\in \mathbb {N}$) a sequence of mappings. We say that the sequence of mappings converges pointwise if for every ${}x\in T$ the sequence

${\left(f_{n}(x)\right)}_{n\in \mathbb {N} }$

converges in ${}M$.