We can construction a sequence of functions in more than one way. We can define a sequence of points
and then define a sequence to be
but this is only actually a sequence of points since each f(x-n) is just a point and not a true function. A true function from which we may construct a sequence is actually a function of both n and x, where n takes all values in the natural numbers. For example we may define
For each sequence so defined we are especially interested in what happens as
The sequence may converge to a well defined function, or may not converge. Convergence may depend on the set of values of
or on a particular
If
is defined on the interval
then if
as 
and if
for all
We say that
converges to
The sequence converges to different points for different values ofIf we had definedon the interval
then the sequence would not have converged because if
as
If the sequenceconverges to a function
it may converge uniformly or pointwise. If a function converges uniformly then
for all
The sequence
does not converge uniformly since if
then
and
hence
and
Hence
is pointwise convergent but not uniformly convergent to