Where $m$ and $n$ are positive integers, consider the rational number $$r = \frac { m } { 3 } + \frac { n } { 7 } .$$ We are to find $m$ and $n$ such that among all $r$'s satisfying $r < \sqrt { 2 }$, $r$ is closest to $\sqrt { 2 }$. It is sufficient that among all $m$'s and $n$'s which satisfy the inequality $$\mathbf { A } m + \mathbf { B } n < \mathbf { CD } \sqrt { 2 }$$ we find the $m$ and $n$ such that $\mathrm { A } m + \mathrm { B } n$ is closest to $\mathrm { CD } \sqrt { 2 }$. Squaring both sides of (1), we have $$(\mathrm { A } m + \mathrm { B } n ) ^ { 2 } < \mathbf { EFG } .$$ Here, the greatest square number which is smaller than EFG is $\mathbf{HIJ} = \mathbf{KL}^{2}$. So, let us consider the equation $$\mathrm { A } m + \mathrm { B } n = \mathrm { KL } .$$ Transforming this equation, we have $$n = \frac { \mathbf { MN } - \mathbf { O } m } { \mathbf { P } } .$$ Since $n$ is an integer, $\mathbf { MN } - \mathbf { O } m$ is a multiple of $\mathbf { Q }$. Thus, we obtain $$m = \mathbf { R } , \quad n = \mathbf { S } .$$
Where $m$ and $n$ are positive integers, consider the rational number
$$r = \frac { m } { 3 } + \frac { n } { 7 } .$$
We are to find $m$ and $n$ such that among all $r$'s satisfying $r < \sqrt { 2 }$, $r$ is closest to $\sqrt { 2 }$.
It is sufficient that among all $m$'s and $n$'s which satisfy the inequality
$$\mathbf { A } m + \mathbf { B } n < \mathbf { CD } \sqrt { 2 }$$
we find the $m$ and $n$ such that $\mathrm { A } m + \mathrm { B } n$ is closest to $\mathrm { CD } \sqrt { 2 }$.
Squaring both sides of (1), we have
$$(\mathrm { A } m + \mathrm { B } n ) ^ { 2 } < \mathbf { EFG } .$$
Here, the greatest square number which is smaller than EFG is $\mathbf{HIJ} = \mathbf{KL}^{2}$. So, let us consider the equation
$$\mathrm { A } m + \mathrm { B } n = \mathrm { KL } .$$
Transforming this equation, we have
$$n = \frac { \mathbf { MN } - \mathbf { O } m } { \mathbf { P } } .$$
Since $n$ is an integer, $\mathbf { MN } - \mathbf { O } m$ is a multiple of $\mathbf { Q }$.\\
Thus, we obtain
$$m = \mathbf { R } , \quad n = \mathbf { S } .$$