In the induction proof for $(1+p)^n \geq 1 + np$, a term is dropped and I don't understand why. In What is Mathematics, pg. 15, a proof of 
$(1+p)^n \geq 1 + np$, for $p>-1$ and positive integer $n$
goes as follows:


*

*Substitute $r$ for $n$, then multiply both sides by $1+p$, obtaining:
$(1+p)^{r+1}\geq 1+rp+p+rp^2$

*"Dropping the positive term $rp^2$ only strengthens this inequality, so that
$(1+p)^{r+1}\geq 1+rp+p$, which shows that the inequality will hold for $r+1$."
I don't understand why the $rp^2$ term can be dropped -- if we're trying to prove that the inequality holds, and dropping $rp^2$ strengthens the inequality, then why are we allowed to drop it?
Thanks!

In $1.$ we have shown that $$(1+p)^{r+1}\geq 1+rp+p+rp^2$$
But we also know that $r > 1$ (because we're doing an induction proof from $1$ upwards); and obviously $p^2 \ge 0$ (because $p$ is real); so we know that $rp^2 \ge 0$. Therefore
$$1+rp+p+rp^2 \ge 1+rp+p$$
So putting these two together gives
$$(1+p)^{r+1}\geq 1+rp+p$$
as required.
In short, if we know that $a \ge b + c$, and we know $c$ is non-negative, we can immediately conclude that $a \ge b$.
