Question

Prove that if h > - 1 , then $1+nh\le (1+h{)}^n$for all nonnegative integers n. This is called Bernoulli’s inequality.

Short answer

$1+nh\le (1+h{)}^n$for all nonnegative integers n

Step-by-step solution

Step: 1

Let P(n) be$1+nh\le (1+h{)}^n$

If n=0,

$\begin{array}{r}1+0h\le (1+h{)}^0\\ 1\le 1\end{array}$

It is true for n=0.

Step: 2

Let P(k) be true.

$1+kh\le (1+h{)}^k$

We need to prove that p(k+1) is true.

Step: 3

$\begin{array}{r}(1+h{)}^{k+1}=(1+h{)}^k\cdot (1+h)\\ \ge (1+hk)\cdot (1+h)\\ =1+hk+h+h^{2}k\\ 1+h(k+1)+h^{2}k\\ \ge 1+(k+1)h\end{array}$