Let . Is uniformly convergent on the interval [0,1]?
I don't think that it is uniformly convergent but I'm having a hard time proving it. I thought to use the theorem that it will be uniformly convergent iff , but I can't see how to show this. Any help?
Moreover, no subsequence of is uniformly convergent on . To see this recall that a uniformly convergent sequence of functions on a compact metric space is necessarily equicontinuous. That said, it's evident that is not equicontinuous. Indeed, let be any subsequence of suppose that there existed such that implies . Note though that the period of is . Thus, since we may choose large enough so that . Thus, since runs through a full period on there exists such that and and so ...but since and (since they're contained within ) this contradicts our choice of . Thus, no such exists and so is not equicontinuous on . It follows then from previous discussion that is not uniformly convergent on .
Moreover, it might be feasible to prove that no subsequence of has a pointwise convergent subsequence on . The proof for why it cannot possess a pointwise convergent subequence on is easy...I'll see if I can adapt my proof.