2. Find all real solutions of the system
$$
\left\{\begin{array}{l}
\frac{4 x^{2}}{1+4 x^{2}}=y, \\
\frac{4 y^{2}}{1+4 y^{2}}=z, \\
\frac{4 z^{2}}{1+4 z^{2}}=x
\end{array}\right.
$$
and prove that your solution is correct.
\%. Let $x=y$. Then $x=y=z$.
From $4 x^{2}=x+4 x^{3}$, we get $x=0$ or $\frac{1}{2}$.
Let $x>y$. Then
$1-\frac{1}{1+4 x^{2}}>1-\frac{1}{1+4 x^{2}}$ or $z>x$,
and $1-\frac{1}{1+4 y^{2}}>1-\frac{1}{1+4 z^{2}}$ or $y>z$.
But $x>y>z>x$ is impossible.
When $x<y$, a similar contradictory statement is obtained.
Therefore, the system of equations has only two solutions:
$(x, y, z)=(0,0,0)$ and $(x, y, z)=\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right)$.