Question about stating propositions in mathematical induction problems. Assume I have a proposition $P(n)$, which states "For a natural number $n$, $n = a + b$ for some natural numbers $a,b$." If I want to prove this proposition only for $n \geq c$, where $c$ is some natural number, should I change my proposition to be "For a natural number $n \geq c$, $n = a + b$", or can I just leave it like so?

I believe you would need to change it because your original proposition has no restrictions on a. This would also change your induction hypothesis, as well as your proof.
