More studying of sequences and series of functions. These boogers are beginning to be a little more manageable.
Problem Statement: Consider the sequence of functions for . (a) Prove converges. (b) Is the convergence uniform? Prove your answer.
(a) I claim that converges pointwise to 0.
Proof:Let and consider
Since as it follows that as since and x is fixed. Thus we may conclude that as for . If then and so . Thus, for it follows that converges pointwise to .
(b) I claim that the convergence is uniform.
Proof: To show uniform convergence we wish to show that the uniform norm goes to 0. Note that since we showed in (a) that the sequence converges to 0 we really only want to show that goes to 0.
Let us consider for and a fixed .
So there is only one zero of and it occurs when . Checking points on either side of tells us that obtains a maximum when . Futhermore we can see that this maximum is . Since each for every and we have shown that each of the are bounded. Moreover, we have found that goes to 0 as since as .
A sequence of bounded functions converges uniformly on a domain A if and only if the uniform norm on A goes to zero, thus we have shown that the sequence converges to uniformly for
Reflection: This was a great problem! I feel like I learned so much in working through this question. It always a little frustrating when they ask you to decide if something converges uniformly or not. At least in a test situation it is. Why can’t they just tell me what to prove?! 🙂 The way I went about doing this problem was a kind of round-about way, but in doing so it led me to better understand this idea of uniform norm. When I learned uniform convergence I was taught to take the derivative of the difference and find where it’s maximum was. Then to show that the difference went to zero as . We never called it a uniform norm, we just did it. I always felt a little shaky about it though. Today I went about solving this problem in the same way and then something clicked. The light went on and suddenly the idea of a uniform norm and what I had been taught meshed together to make this super idea. What I had been doing all along was showing that the uniform norm goes to zero! I just never knew it. So now, after having discussed and chewed the fat of this problem for quite a while I am feeling much better about uniform convergence.
Only 25 (eh, it’s 10pm, let’s call it 24) days until the exam. I think I can, I think I can, I think I can…