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)
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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.
Take a look at Chapter 1, sections 11 and 12 of Tauberian Theory: A Century of Development by Jacob Korevaar. He also has many references.