For any positive integer $m$,show that there exist integers $a,b$ satisfying $\left | a \right |\leq m$, $ \left | b \right |\leq m$, $0< a+b\sqrt{2}\leq \frac{1+\sqrt{2}}{m+2}$
Is there an intuitive approach to this? It's a question from the book art and craft of problem solving dealing with the pigeonhole principle.
It seems like one of those problems that is really hard to get any information out of. If both conditions $|a|\le m, |b|\le m$ are to be satisfied at the same time, then taking the absolute value of the expression $0< a+b\sqrt{2}\leq \frac{1+\sqrt{2}}{m+2}$ one has that $$|a|+|b|\sqrt2 \le \frac{1+\sqrt{2}}{m+2}$$ but we also have that $$|a|+|b|\sqrt2 \le m+m\sqrt2 =m(1+\sqrt2)$$ which has the same term $1+\sqrt2$ as does the given fraction in it's numerator, but I don't think I can draw any other information out of this. What kind of considerations should be done here?