Example 2 (1996 National High School League Question) The number of proper subsets of the set $\left\{x \left\lvert\,-1 \leqslant \log _{\frac{1}{x}} 10<-\frac{1}{2}\right., x \in \mathbf{N}_{+}\right\}$ is $\qquad$ .
Solve: Fill in $2^{90}-1$. Reason: By the change of base formula, $-1 \leqslant \log _{\frac{1}{x}} 10<-\frac{1}{2} \Leftrightarrow-1 \leqslant \frac{1}{\lg _{\frac{1}{x}}}<-\frac{1}{2} \Leftrightarrow$ $1 \leqslant \lg x<2 \Leftrightarrow 10 \leqslant x<100$. Since $x \in \mathbf{N}_{+}$, we have $x=10,11,12, \cdots, 99$, a total of 90 different values, so the number of proper subsets is $2^{90}-1$.