5. Let for any natural numbers $m$, $n$ satisfying $\frac{m}{n}<\sqrt{7}$, the inequality $7-\frac{m^{2}}{n^{2}} \geqslant \frac{\lambda}{n^{2}}$ always holds. Then the maximum value of $\lambda$ is $\qquad$.
5.3.
The original inequality $\Leftrightarrow 7 n^{2}-m^{2} \geqslant \lambda$.
Since $7 n^{2} \equiv 0(\bmod 7), m^{2} \equiv 0,1,2,4(\bmod 7)$, then $\left(7 n^{2}-m^{2}\right)_{\text {min }}=3$, so, $\lambda_{\text {max }}=3$.