I found the following question in a book only with one sentence. "This question can be solved by an elementary way. Note that the following two are false: (1) If a function is bounded from below, then it has minimum value. (2) A monotone decreasing sequence reaches a negative value."
Question: Let $m,n$ be integers. Supposing that a function $f(m,n)$ defined by $m,n$ satisfies the following two conditions, then prove that $f(m,n)$ is a constant.
I suspect this question can be solved by a geometric aspect. I've tried to prove this, but I'm facing difficulty. Could you show me how to prove this?