Example 10 The real number $a$ makes the inequality
$$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2} \geqslant a\left(x_{1} x_{2}+x_{2} x_{3}+x_{3} x_{4}+x_{4} x_{5}\right)$$
hold for any real numbers $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$. Find the maximum value of $a$. (2010 Shanghai High School Mathematics Competition)
Solve for the maximum value of $a$ which is $\frac{2 \sqrt{3}}{3}$.
Because when $x_{1}=1, x_{2}=\sqrt{3}, x_{3}=2, x_{4}=\sqrt{3}, x_{5}=1$, we get $a \leqslant \frac{2}{\sqrt{3}}$.
Furthermore, when $a=\frac{2}{\sqrt{3}}$, the inequality always holds.
In fact,
$$\begin{aligned}
x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2} & =\left(x_{1}^{2}+\frac{x_{2}^{2}}{3}\right)+\left(\frac{2 x_{2}^{2}}{3}+\frac{x_{3}^{2}}{2}\right)+\left(\frac{x_{3}^{2}}{2}+\frac{2 x_{4}^{2}}{3}\right)+\left(\frac{x_{4}^{2}}{3}+x_{5}^{2}\right) \\
& \geqslant \frac{2}{\sqrt{3}} x_{1} x_{2}+\frac{2}{\sqrt{3}} x_{2} x_{3}+\frac{2}{\sqrt{3}} x_{3} x_{4}+\frac{2}{\sqrt{3}} x_{4} x_{5}
\end{aligned}$$

Therefore, the maximum value of $a$ is $\frac{2 \sqrt{3}}{3}$.