Example 1 Find the positive integer root of the equation
$$
\frac{2}{n}+\frac{3}{n+1}+\frac{4}{n+2}=\frac{133}{60}
$$

(1990, Shanghai Junior High School Mathematics Competition)
Solution: Since $n$ is a natural number, by the properties of natural numbers and
$$
\begin{array}{l} 
\frac{1}{n}>\frac{1}{n+1}>\frac{1}{n+2} \\
\therefore \frac{2+3+4}{n+2}<\frac{2}{n}+\frac{3}{n+1}+\frac{4}{n+2} \\
<\frac{2+3+4}{n} .
\end{array}
$$

That is, $\frac{9}{n+2}<\frac{133}{60}<\frac{9}{n}$.
Solving this, we get $2 \frac{8}{133}<n<4 \frac{8}{133}$.
Since $n$ is a natural number, $n$ can only be 3 or 4. Upon verification, $n=3$ is the positive integer solution of the equation.