Proposition: Bernoulli Inequality
Proof:
The proof goes by induction.
- Property Q(n):
- If x -1 then
(1 + x) n 1 + nx
- Check Q(1):
- (1 + x) 1 + x is true.
- Assume Q(n) is true:
- If x -1 then
(1 + x) n 1 + nx
- Check Q(n+1):
-
(1 + x) n + 1 = (1 + x) n (1 + x) (1 + nx) (1 + x) =
so Q(n+1) is true (where have we used that x -1 ?).
1 + nx 2 + nx + x = 1 + nx 2 + (n + 1)x 1 + (n + 1)x