There is no basis in deductive logic for inductive proofs. No deductive proof exists that induction works. It is something we take on faith, just like we take axioms and predicates on faith. Induction, like the "like" operator, (with which it has much in common), does not have a corresponding logical tautology, nor a proved theorem to it's name. So your example will not help you. I asked you to combine lemmas from the two disciplines to produce a valid proof. When you are talking about lemmas, you are usually engaged in formally serious efforts, and therefore talking about deductive, not inductive proof. Inductive proof is handwaving over the notion that what you have learned to expect is what you expect. It does not provide formal deductive security to a theorem.
Nicely put. I'll bow to your explanation, which sounds about right to me. There's indeed an implicit assumption that no counter-examples can be found -- which is a reasonable assumption in certain discrete cases. Inductive proofs are not necessarily true in non-discrete cases -- to my pea brain the Mandelbrot gives a good visual example of this.
(I'm also not interested in going to great lengths to prove my off-the-cuff comment....)
It's interesting to note, BTW, that your thumbnail description of inductive proofs is a fair description of how we go through life.
Those who claim to have derived a perfectly rational and logical ethical system are essentially claiming to do so on an inductive basis (they can only go by what they observe, and cannot have addressed all possible cases). Even the inductive claim is not convincing, given that there are observable counterexamples. The deductive proof is obviously out of the question.