Example 3 The number of proper subsets of the set $\left\{x \left\lvert\,-1 \leqslant \log _{\frac{1}{x}} 10<-\frac{1}{2}\right., x\right.$ $\left.\in \mathbf{N}_{+}\right\}$ is $\qquad$ .
(1996, National High School Mathematics Competition)
$$
\begin{array}{l}
\text { Solution: } A=\left\{x \left\lvert\,-1 \leqslant \log _{\frac{1}{x}} 10<-\frac{1}{2}\right., x \in \mathbf{N}_{+}\right\} \\
=\left\{x \left\lvert\,-1 \leqslant-\log _{x} 10<-\frac{1}{2}\right., x \in \mathbf{N}_{+}\right\} \\
=\left\{x \mid 1 \leqslant \lg x<2, x \in \mathbf{N}_{+}\right\} \\
=\left\{x \mid 10 \leqslant x<100, x \in \mathbf{N}_{+}\right\} .
\end{array}
$$

For a set, if the set has $n$ elements, then the number of its subsets is $\mathrm{C}_{n}^{0}+\mathrm{C}_{n}^{1}+\cdots+\mathrm{C}_{n}^{n}=2^{n}$, and the number of proper subsets is $2^{n}-1$. Therefore, set $A$ has 90 elements, and has $2^{90}-1$ proper subsets.
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