how do I prove this inequallity? [duplicate]
$m,n$ are integers bigger than 0.
I need to prove:
$$\frac{1}{\sqrt[n]{1+m}}+\frac{1}{\sqrt[m]{1+n}} \geq 1$$
I tried to multiply both sides by the common factor and raise both sides to the power of $m*n$ but it did not work for me, and I have no other idea how to proceed.
I don't want the full solution, but just a hint.
Thank you.
HINT
By Bernoulli's inequality we have
$(1+m)^\frac1n \le 1+\frac m n$
$(1+n)^\frac1m \le 1+\frac n m $