Proof by Strong Induction

Used for proving sequences and series. WTS: $\forall k\ge b,(\forall b\le i<k,P(i))⟹P(k)$

1. Suppose arbitrary $k$
2. Suppose $k\ge$ smallest base case
3. Suppose base case 1:
   1. …
   2. P(1)
4. Suppose base case 2:
   1. …
   2. P(2)
5. Suppose $k\ge b$
   1. Suppose $\forall b<i<k,P(i)$
   2. …
   3. $P(k)$

Proof Strategy

You need to prove manually $P(b),P(b+1),\dots$ (depends on what you need to use)