TITLE: The limit of the nth root of a to the n plus b to the n is the maximum of (a,b)
QUESTION [6 upvotes]: I've been asked to prove the following from Spivak's Calculus
$$\lim_{n\to\infty}\sqrt[n]{a^n+b^n}=\max(a,b); a,b > 0$$
I understand that this is a proof by cases, and that our cases are $a=b$, $a>b$, and $b>a$. I have done the $a=b$ case, but I am stuck on the $a>b$ and $b>a$ cases.
Some hints would be appreciated-
Thanks!

REPLY [3 votes]: we assume $a<b$.
Now use the sandwich to obtain: 
$b=\lim\limits_{n\to \infty} \sqrt[n]{b^n}\leq\sqrt[n]{a^n+b^n}\lim\limits_{n\to \infty} \sqrt[n]{a^n+b^n}\leq \lim\limits_{n\to \infty} \sqrt[n]{2b^n}=\lim\limits_{n\to \infty}\sqrt[n]{2}\lim\limits_{n\to \infty}\sqrt[n]{b^n}=1b$

REPLY [2 votes]: Hint: 
If $a \leqslant b$
$$b \leqslant (a^n + b^n)^{1/n} \leqslant 2^{1/n} b$$