Supremum and infimum of $\{x, y \ge 1 : \frac{xy}{3x + 2y + 1}\}$

Question

How to find supremum and infimum of $\{x, y \ge 1 : \frac{xy}{3x + 2y + 1}\}$? I suspect that $\frac{1}{6}$ is infimum and supremum does not exist, but I dont know how to prove it using only the definition of sup/inf.

Answer 1

Observe it as $$f(x):= \frac{xy}{3x + 2y + 1}$$ with parameter $y$. It is easy to see that $f$ is increasing for $x\geq 0$ (draw a graph). So $$f(x) \geq f(1)= {y\over 2y+4} =: g(y)$$

But $g$ is also increasing for $y> -2$ so $$g(y) \geq g(1) ={1\over 6}$$ So this infimum is actualy minumum.