Thread: I got to figure this out tonight

please help.

I need to find a uniformly convergent sequence of functions on (0,1) whose derivatives do not converge.

then i need to find a convergent sequence of integrable functions whose limit is not integrable

im thinking fn = 1/nsinx for the first one and for the second I don't really have a clue.

Originally Posted by Smancer

please help.

I need to find a uniformly convergent sequence of functions on (0,1) whose derivatives do not converge.

then i need to find a convergent sequence of integrable functions whose limit is not integrable

im thinking fn = 1/nsinx for the first one and for the second I don't really have a clue.

For the second, let rn be the countable rational numbers in the interval [0,1] and define the functions.

gn(x)= {1,0 if x= r1,r2,r2,... rn and g(x) = 1 if x = irrational and g(x)=0 if x=rational

then g(n) converges point wise to g
the limit function is not integrable

Does this help or are you looking for something else?

I think I get it... thanks I was trying to build small intervals around the rational numbers and have them shrink as n approached infinity. I couldn't quite think of a way to do that though.

Any ideas on the first?