Hello everyone, first post! I've come across the following Analysis problem and given that it's one of the hardest questions on my problem sheet, I am having a few problems working my argument through.
"Suppose that f_n:[0,1] -> Reals is a sequence of continuous functions tending pointwise to 0. Must there be an interval on which f_n -> 0 uniformly?"
I have considered using the Weierstrass approximation theorem here, which states that we can find, for any continuous function [0,1] -> Reals, a uniform approximation by polynomials.
Because of this, it seems to me - though I could be wrong - that these f_n -> 0 uniformly if this series of polynomials (each p_n approximating an f_n to a sufficient degree of accuracy) tends to 0 uniformly - in which case it suffices to prove the result for any series of polynomials.
Even if this deduction -is- correct, which I'm not 100% confident about, I can't seem to follow through and show that there exists such an interval for a polynomial sequence. On the other hand, perhaps there is a counterexample and I'm going about this completely the wrong way! Could anyone lend a hand please?