Example 10 Find all real solutions of the system of equations
$$
\left\{\begin{array}{l}
\frac{4 x^{2}}{1+4 x^{2}}=y, \\
\frac{4 y^{2}}{1+4 y^{2}}=z, \\
\frac{4 z^{2}}{1+4 z^{2}}=x
\end{array}\right.
$$

and prove that your solution is correct. (1996, Canadian Mathematical Olympiad)
Solution: Clearly, $x=y=z=0$ is a solution to the system of equations. If $x, y, z$ are all non-zero, taking the reciprocal yields
$$
\left\{\begin{array}{l}
\frac{1}{4 x^{2}}-\frac{1}{y}+1=0, \\
\frac{1}{4 y^{2}}-\frac{1}{z}+1=0, \\
\frac{1}{4 z^{2}}-\frac{1}{x}+1=0 .
\end{array}\right.
$$

Adding them up gives $\left(\frac{1}{2 x}-1\right)^{2}+\left(\frac{1}{2 y}-1\right)^{2}+\left(\frac{1}{2 z}-1\right)^{2}=0$.
Since $x, y, z$ are real numbers,
$$
\therefore \frac{1}{2 x}-1=\frac{1}{2 y}-1=\frac{1}{2 z}-1=0 \text {. }
$$

Thus, $x=y=z=\frac{1}{2}$.
Therefore, the solutions to the original system of equations are $x=y=z=0$ or
$$
x=y=z=\frac{1}{2} \text {. }
$$