On , the function sequence converge uniformly to respectively. Suppose there exists positive sequence such that . Prove that converge unformly to on

PS: If , I know how to prove. But this?....Would you help me?

Printable View

- Dec 18th 2012, 04:40 PMxinglongdadaconverge uniformly?
On , the function sequence converge uniformly to respectively. Suppose there exists positive sequence such that . Prove that converge unformly to on

PS: If , I know how to prove. But this?....Would you help me? - Jan 2nd 2013, 10:51 PMMacstersUndeadRe: converge uniformly?
To show that converges uniformly to on , we have to show there exists an N such that for all x in the interval and ,

I'm sorry I'm a bit rusty, but perhaps it would help to note that you can consider an upper bound for your positive sequence . If the least upper bound for is a real number, say M, then you can say and for all n. If there is no least upper bound for your positive sequence, I'm not sure how to continue. - Jan 2nd 2013, 11:53 PMhollywoodRe: converge uniformly?
So what you're saying is that and are sequences of bounded functions that converge uniformly to and respectively on , and you want to prove that converges uniformly to .

You said you know how to do it if and are uniformly bounded - that is, there are and such that on and on . So prove that first:

Let be such that for all . Then for all :

and I think you can probably fill in the rest of the proof.

- Hollywood - Jan 3rd 2013, 05:29 PMxinglongdadaRe: converge uniformly?
Thank you very much indeed.