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Let $a\sqrt{2} = m + \alpha$ where $m$ is an integer and $0 < \alpha < 1$, and $b\sqrt{7} = n + \beta$ where $n$ is an integer and $0 < \beta < 1$. (The fractional parts are positive since $\sqrt{2}$ and $\sqrt{7}$ are irrational.) We are to prove that $$(m+ \alpha)(n + \beta) - mn \geq 1.$$ We have \alpha = a\sqrt{2} - m = \frac{2a^2 - ...