Theorem
The limit of a uniformly convergent sequence of bounded mappings is bounded.
Proof
Let
where
is an arbitrary set and
is a metric space. The sequence
converges uniformly to
if and only if
Consider the space
of all bounded mappings
with the metric defined by
We have
Hence in the spacethe condition
means that the sequence
converges uniformly to
Let
and let
whereis a uniformly convergent sequence of bounded mappings. Choose
such that
We have
Henceis bounded.