I couldn't find a similar question, so here goes: I would like to prove the fact (?) that the sequence of functions $(f_n) \subset C([0,1])$ defined by $f_n(x)=\sin(nx)$ does not have a subsequence that converges pointwise to any function.
This is not homework. Soon I am going to teach the Arzela-Ascoli theorem, and I want to show that without the equicontinuity assumption, it is difficult to conclude anything about convergence of a sequence of a functions, even pointwise convergence.
I am almost certain that what I assert above is true, I am just having trouble proving it. I am also almost certain that this has been done before. It is obvious that the entire sequence $(f_n)$ doesn't converge pointwise, and that no subsequence of $(f_n)$ converges uniformly, but that is not my question.