Example 3 (2000 Hebei Provincial Competition Question) Given $a, b \in \mathbf{R}^{+}, m, n \in \mathbf{R}, m^{2} n^{2}>a^{2} m^{2}+b^{2} n^{2}$, let $M=\sqrt{m^{2}+n^{2}}, N=a+b$, then the size relationship between $M$ and $N$ is ( ).
A. $M>N$
B. $M<N$
C. $M=N$
D. The size relationship between $M$ and $N$ cannot be determined