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…

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Erin – I really like the proof. Great job! My only suggestion would be to change the line “Checking points on either side… obtains a maximum” to say local maximum rather than just maximum. The subsequent line argues that the local maximum itself is a global maximum (with respect to your domain), and so serves as a proper upper bound of your sequence. Protect those 5 points!!!