If we have a sequence of functions that converges uniformly on , and each is continuous on E, is the limit function
Let's fix . Since we know that the sequence of functions is uniformly convergent on E, we can find an integer N so that implies
for all .
Additionally, we know that each of these functions is continuous, so for all we can find a so that
if for all in .
Then the following inequality
shows that for ,
Therefore, we conclude that the limit function is continuous on under the following conditions.