Infimum of the set $\{(1/m)^{1/n}+(1/n)^{1/m}\mid m,n\in\Bbb N\}$

Question

I have to find an infimum of the set $A:=\left\{\frac{1}{\sqrt[m]{n}}+\frac{1}{\sqrt[n]{m}}: m,n\in\mathbb{Z}_+\right\}$. I think that it is $1$. It is easy to find a sequence of numbers from $A$ that converges to $1$ (set $m=n^n$). But I don't know how to show that $a\geqslant 1$ for $a\in A$.