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

Therefore, the number of its proper subsets is
$$
\mathrm{C}_{90}^{0}+\mathrm{C}_{90}^{1}+\cdots+\mathrm{C}_{90}^{89}=2^{90}-1 .
$$