In problem 3, where f is supposed to be defined? You deal with the limit at infinity, but f seems to be only defined on [0,1).
problem 3:
Prove that if f : [0,1)--- IR is nonnegative, integrable, and uniformly
continuous, then lim f(x) =0 as x tends to 00.
Problem 4:
Suppose that a differentiable function f : IR ---IR and its derivative f'
have no common zeros. Prove that f has only finitely many zeros in [0, 1].
For the first one, assume that it doesn't, then there exists and a sequence such that and . By uniform continuity there is a such that if then then if satisifies this we have, by the triangle inequality but then
for all . This is a contradiction.
For the second one, since is compact, assuming there is a sequence of zeroes of , we get that there is a subsequence that converges to something in and this limit is also a zero. Using Rolle's theorem we can construct a sequence of zeroes of that converges to the same limit point. If were continous this would be enough, but at the moment I don't see how to finish the argument in general.