You don't have to. See my top level comment. In short, you have to strengthen your goal in order to have a strong enough inductive hypothesis to prove anything.

You don't have to. See my top level comment. In short, you have to strengthen your goal in order to have a strong enough inductive hypothesis to prove anything. E.g., instead of "picking a base case" that's sufficient, you work the base case into the goal, like n >= 2 -> P(n). Then you do the base case as part of a branch in the inductive step, without using your inductive hypothesis.