Proposition: Bernoulli Inequality

If x -1 then (1 + x) n 1 + nx for all positive integers n.
Context Context

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) =
1 + nx 2 + nx + x = 1 + nx 2 + (n + 1)x 1 + (n + 1)x
so Q(n+1) is true (where have we used that x -1 ?).
As an alternative proof, this statement follows directly from the binomial theorem (do it !).

Interactive Real Analysis, ver. 1.9.5
(c) 1994-2007, Bert G. Wachsmuth
Page last modified: Mar 26, 2007