x belongs to [0,1]
limit n->infinity (limit m-> infinity cos ^{2n} (m! pi x) )=?
its value is 0 if x is irrational
and it takes 1 if x is rational.
please explain how it happens.
I am not sure about this nested limit, but we get the Dirichlet function if we swap limits:
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If x is rational, then m!x is eventually an (even) integer. If x is irrational, then , so the inner limit is 0.