Definition: Let
be a subset of
and let
be a sequence of functions defined on
We say thatconverges pointwise onif 
exists for each point
This means thatis a real number that depends only on
Ifis pointwise convergent then the function defined by
for every
is called the pointwise limit of the sequence
More formally, given any
and 
there exists a natural number
depending only on
and
such that
for every
Definition Letbe a subset ofand letbe a sequence of real valued functions defined onThenconverges uniformly to
if given anythere exists a natural number
such thatfor every
depends only on so uniform convergence implies pointwise convergence., but the converse is not true.
Example:
As
for eachso converges pointwise to zero.
Suppose now that
and
then
and
sois not uniformly convergent.