Solving an inequality with varying integer exponents

Question

Do there exist positive integers $n,m$ such that

$$1>r^n+r^m-r^2r^nr^m$$

for all $r \in (0,1)$? If so, what are the smallest such pairs?

Answer 1

This is equivalent to

$$r^2 r^nr^m - r^n - r^m > -1$$

$$r^4 r^n r^m - r^2 r^n - r^2 r^m > -r^2$$

$$(r^2 r^n - 1)(r^2 r^m - 1) > 1-r^2$$

$$(1-r^{2+n})(1-r^{2+m}) > (1-r^2)$$

Looking at the function $f_r(k) = 1-r^k$, can you see why this is never true?