Problem Statement: For every and define . Find the function that converges to pointwise. Is the converges uniform?
Proof: Fix . Then and so converges to pointwise.
Claim: This convergence is not uniform.
If the convergence is uniform then it follows that for every we may find an such that for every and for every . Let and . Since this implies that . This is clearly a contradiction and so we have found a counterexample such that the sequence breaks uniform continuity. Thus, converges to pointwise but not uniformly.
Reflection: This one tripped me up a bit. I was able to get the piecewise convergence but I could not decide whether it converged uniformly or not. My intuition told me that it didn’t, but I couldn’t think of a counter example like above where it broke uniform convergence. While trying to find a counter example I saw that as got closer to the needed to make converge got larger and larger. This made me think that we couldn’t find a universal that would work for all . After seeing the above counter example it makes a lot more sense. I think what I need to keep in mind when trying to find a counter example for these kind of things is that I don’t necessarily want to find a specific value where it breaks, but that I should tailor my value to what I know, i.e. the .