Prove: $(a + b)^{n} \geq a^{n} + b^{n}$ 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...

Induction.
For $n=1$ it is trivially true
Assume true for $n=k$
i.e. $$(a + b)^k \ge a^k + b^k$$
Consider case $n=k+1$
\begin{align}&(a+b)^{k+1}
=(a+b)(a+b)^k\\
&\ge(a+b)(a^k+b^k)\\
&=a^{k+1}+b^{k+1}+ab^k+ba^k\\
&\ge a^{k+1}+b^{k+1}\end{align}
