For $a<b<c$, is it true that $c^n>a^n+b^n$ for large enough $n$?

Question

Given three numbers $a$, $b$, $c$ with $a < b < c $, does the following hold?

$$c^n > a^n+b^n \qquad \text{ for large enough }n$$

Answer 1

Yes, if $a\ge0$. For then we use $$0\le\frac ac\le \frac bc<1.$$ Hence $\left(\frac ab\right)^n<\frac12$ and $\left(\frac bc\right)^n<\frac12$ for $n\gg0$.

If we allow $a<0$, consider $a=-2,b=0, c=1$, which makes the inequality false for infinitely many (namely all even) $n$.