TITLE: Prove: $(a + b)^{n} \geq a^{n} + b^{n}$
QUESTION [10 upvotes]: Struggling with yet another proof:

Prove that, for any positive integer $n: (a + b)^n \geq a^n + b^n$ for all $a, b > 0:$

I wasted $3$ pages of notebook paper on this problem, and I'm getting nowhere slowly. So I need some hints.
$1.$ What technique would you use to prove this (e.g. induction, direct, counter example)
$2.$ Are there any tricks to the proof? I've seen some crazy stuff pulled out of nowhere when it comes to proofs...

REPLY [12 votes]: You can write $n=m+1$ where $m \geq 0$, then
$(a+b)^n = (a+b)^{m+1} = (a+b) (a+b)^m = a(a+b)^m +b(a+b)^m \geq a^{m+1} + b^{m+1}$
no induction and works for any real $n \geq 1$.