The important thing, is that the output of a non-linear device depends only on the current input. That means, that if the input repeats, so does the output.
If the input is periodic with the period T, then the output must also be periodic with the period T. But that means, that it can only be the fundamental and it’s harmonics (they all repeat with a period T).
You may also study, how simple powers (x^(n)) affect the frequencies: they create scaled copies at sums and differences of every two frequencies of the input signal (including the negative ones). Now remember, that many nonlinear functions can be decomposed into power series (called Taylor series) – so they are just a scaled sum of powers.
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