Prove that all for all positive integers $m,n $: $\frac{1}{\sqrt[n]{m}}+\frac{1}{\sqrt[m]{n}}>1$

Question

I am aware of the famous result of $x^{y}+y^{x}>1$ and its proof via Bernoulli's inequality, so I presumed this would be a simple application with minor tweaks, but I have not been successful in deriving the result from the question. Should one proceed from Bernoulli in a different way or use a completely different method altogether? Thank you!

Answer 1

Turning my earlier comment into an answer: The result follows by setting $x = \frac1{m}$ and $y = \frac1{n}$ in the inequality $x^y + y^x > 1$.