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?
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.
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.