Example 2 Let $A$ be a subset of the set of positive integers $\mathbf{N}^{*}$. For any $x, y \in A, x \neq y$, we have $\mid x- y \left\lvert\, \geqslant \frac{x y}{25}\right.$. Find the maximum value of $|A|$. (26th IMO Shortlist)
$$\begin{array}{l}
\text { Let } X_{1}=\{1\}, X_{2}=\{2\}, X_{3}=\{3\}, X_{4}=\{4\}, X_{5}=\{5,6\}, \\
X_{6}=\{7,8,9\}, X_{7}=\{10,11, \cdots, 16\}, X_{8}=\{17,18, \cdots, 53\}, X_{9}=\{54, \\
55, \cdots\}=\mathbf{N}^{*} \backslash\{1,2, \cdots, 53\} .
\end{array}$$

For $X_{9}$, when $x, y \in X_{9}$, $x>25$, so $y-x<y<y \cdot \frac{x}{25}=\frac{x y}{25}$. And for $\dot{X}_{i}(i=1,2, \cdots, 8)$, when $x, y \in X_{i}$, it is clear that $y-x<\frac{x y}{25}$. Therefore, $A$ can contain at most one number from each of the above sets, so $n \leqslant 9$.

Furthermore, $A=\{1,2,3,4,5,7,10,17,54\}$ satisfies the conditions, so the maximum value of $|A|$ is 9.