Uniform convergence in the Helly selection principle

Jan 23, 09:41 PM
At the end of today’s discussion I showed you that we could get uniform convergence on the real line in Helly’s selection principle if the limit function was continuous. Unfortunately, the proof I gave was wrong. Furthermore the result is wrong!

Here is an example where uniform convergence fails:

Let Pasted Graphic 8 if Pasted Graphic 10 and Pasted Graphic 9 if Pasted Graphic 11. Clearly Pasted Graphic 12 is monotonic for each Pasted Graphic 13 andPasted Graphic 17 for all Pasted Graphic 15. However the convergence is not uniform although the limit function is certainly continuous!

Note however that Pasted Graphic 16 does converge uniformly to 0 on compact sets. (This explains the difference between the different printings of the book. The problem as stated in my book is wrong!)

I’ll leave you the challenge for now of seeing where my “proof” went wrong. We’ll talk about this on monday.