A question on sequence of functions.

[MathematicalPhysicist]

ok, what about the question with which iv'e opened the thread?

 

[Dick]

i thought a periodic function has a small period which is non zero, here clearly there isn't such period.
anyway, if this is considered a periodic function then back to the original question is there such a function which is not periodic?

I think NateTG's example sin(x)+cos(pi*x) is an excellent example of a function that statisfies your requirements (as are 'almost periodic' functions in general). How many times has that been stated on this thread? Can you prove it?

 

[MathematicalPhysicist]

i think by arzela ascoli theorem i only need to show that it's uniformly bounded (obviously it is by: 2. and that this sequence of functions is equicontinuous, which is also bacause of the first derivative is uniformly bounded so it's equicontinuous.

ok, thanks Dick,Nate and also Matt, i appreciate your help on this question.