My prof. just taught us the method of mathematical induction today, and I'm still a little confused on the "Basis step" of the induction procedure.
Why do we have to first prove that p(1) is true, if $p(n) = 3 \mid(n^4 - n^2)$, for all $n \in \mathbb N$ for example.
doesn't the inductive step: " $3|(n^4 - n^2)$ implies $3|((n+1)^4 - (n+1)^2)$ " already a step that proves $3|(n^4 - n^2)$ is true for all $n \in \mathbb N$? Which includes "$n=1$", if the inductive hypothesis is true (AKA we assume the antecedent is true).
According to my guess, I think that we have to start somewhere and show that the initial value is true, then we can continue the inductive step? However, the Basis step is not a necessary requirement in the proof?