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In the context of Peano arithmetics, mathematical induction is actually an axiom, that is you cannot prove it from other properties of the natural numbers. It is extremely plausible that if something holds for $0$, and if it holds for $n$ it also holds for $n+1$, that it holds for all natural numbers. However the only reason why you can use it in proofs is that you have it as axiom. In other words, it is *assumed* that it works. Therefore I'd argue that it is justified to call it “induction”.