TITLE: Why is this inequality important? $(1+p)^n\geq 1+np$
QUESTION [10 upvotes]: I'm reading Courant's What is Mathematics?
In the beginning, he's showing some proofs, there's a proof about An Important Equality and this important equality is:
$$(1+p)^n\geq 1+np$$
The book's author states the proof but he doesn't say why the inequality is important, I had a feeling that this inequality importance is by stabilishing the $\geq$ relation in $\mathbb{N}$, is this it's importance? If not, what is it?

REPLY [2 votes]: It can be used (as shown in "What is Mathematics") to prove that $n^{1/n} \to 1$ as $n \to \infty$ for integer $n$ by starting with $(1+1/\sqrt{n})^n$. This has been discussed here (by me, naturally), so I'll leave the details for the reader.
It can also be used to prove the arithmetic-geometric mean inequality for the case when $n-1$ of the $n$ items are the same. This, in turn, can be used to show that $\lim_{n \to \infty} (1+1/n)^n$ exists (for integer $n$).