Induction proof: If $h > -1$, then $1 + nh \leq (1 + h)^n$ for all nonnegative integers $n$.

Question

So far I have this:

Prove: $h > -1$, then $1 + nh \leq (1 + h)^n$ for all nonnegative integers $n$.

Base case: $n = 0$

$1 + (0)h \leq (1 + h)^0$

$1 \leq 1$

$1 = 1$

Assume $1 + nh \leq (1 + h)^n$ for all nonnegative integers $n$.

We must show that $1 + (n + 1)h \leq (1 + h)^{n + 1}$

Now I'm stuck. What am I supposed to do now? Thanks for any help.

Answer 1

Hint:

As $1+h>0$,

$(1+h)^{n+1}=(1+h)^n(1+h)\ge(1+nh)(1+h)=1+(n+1)h+nh^2$