Cauchy’s Condition for Uniform Convergence
Specifically, a sequence converges uniformly to a function if and only if for every there exists an so that and imply that .
One direction is straightforward. Assume that converges uniformly to . Given we can pick so that implies that for all . Then if and we have
In the other direction, if the Cauchy condition holds for the sequence of functions, then the Cauchy condition holds for the sequence of numbers we get by evaluating at each point . So at least we know that the sequence of functions must converge pointwise. We set to be this limit, and we’re left to show that the convergence is uniform.
Given an the Cauchy condition tells us that we have an so that implies that for every natural number . Then taking the limit over we find
Thus the convergence is uniform.