Well this one looks very interesting, it looks pretty obvious but I can't find a way to prove it formally:
Let a,b,c be positive numbers.
Prove:
$\displaystyle lim(a^n+b^n+c^n)^\frac{1}{n} = max(a,b,c)$
Let's suppose that $\displaystyle a>b$ and $\displaystyle a>c$ [othervise you can always swap the variables ... ]. Because is...
$\displaystyle \sqrt[n]{a^{n} + b^{n} + c^{n}} = a \sqrt{1+(\frac{b}{a})^{n} + (\frac{c}{a})^{n}}$ (1)
... and...
$\displaystyle \frac{b}{a}<1$ , $\displaystyle \frac{c}{a}<1$ (2)
... from (1) and (2) we derive...
$\displaystyle \lim_{n \rightarrow \infty} \sqrt[n]{a^{n} + b^{n} + c^{n}} = a$ (3)
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$