How can I prove the convergence of the sequence $b_n=\sqrt[n]{x^n+y^n}$ where $x, y > 0$?
Can I divide it in two cases?
Case 1: $x > y$.
$$ b_n=\sqrt[n]{x^n+y^n} < \sqrt[n]{x^n+x^n} = \sqrt[n]{2 \cdot x^n}=x \cdot \sqrt[n]{2} $$
Case 2: $x < y$.
$$ b_n=\sqrt[n]{x^n+y^n} < \sqrt[n]{y^n+y^n} = \sqrt[n]{2 \cdot y^n}=y \cdot \sqrt[n]{2}$$
Result: Does the sequence converges to $\max(x,y)$?