Reading: fa15 notes
Suggestions for when you get stuck on a proof
it's often useful to use the definitions you have, either because you know them or because they're given in the problem
statements to be proven often suggest a proof structure. For example, a proof that \(∀ x, P(x)\) often starts with "choose an arbitrary \(x\)." See the fa15 notes for a systematic list of proof structures.
One way to summarize induction is by the statement:
Induction principle: \(P(0)\) and \(∀n, (P(n) \implies P(n+1))\) together imply \(∀n, P(n)\).
Breaking this down by the rules linked above, to use this prove \(∀n, P(n)\), you can first prove (\(P(0)\) and \(∀n, (P(n) \implies P(n+1))\). Proving \(P(0)\) is of course the base case, while proving \(∀n, (P(n)\implies P(n+1))\) gives the inductive step. How does one prove \(∀n, (P(n) \implies P(n+1))\)? By the table linked above, you start by choosing an arbitrary \(n\). Then, you are trying to prove \(P(n) \implies P(n+1)\) for that \(n\); so you assume \(P(n)\), and your goal becomes showing that \(P(n+1)\) holds.