Proof of the Hardy-Littlewood Tauberian theorem Can someone point me to a proof of the Hardy-Littlewood Tauberian theorem, that is suitable enough to be shown to high school students? (with knowledge of calculus, sequences and series of course)

Have you looked at the presentation in Titchmarsh's Theory of Functions (Section 7.5)? The only non-elementary part of the argument is Weierstrass's approximation theorem, which you can probably assume as a fact. The preliminary material given also include an "easy" special case where the exposition certainly can be understood by someone with knowledge of calculus, sequences, and series.
