1. The number of proper subsets of the set $\left\{x \left\lvert\,-1 \leqslant \log _{\frac{1}{x}} 10<-\frac{1}{2}\right., x \in N\right\}$ is
$$
\begin{array}{l}
\text { II. 1. } 2^{90}-1 . \\
\text { The set }\left\{x \left\lvert\,-1 \leqslant \log _{\frac{1}{x}} 10<-\frac{1}{2}\right., x \in \mathbb{N}\right\} \\
=\left\{x \left\lvert\,-1 \leqslant \frac{1}{\lg \frac{1}{x}}<-\frac{1}{2}\right., x \in \mathbb{N}\right\} \\
=\{x \mid 1 \leqslant \lg x<2, x \in \mathbb{N}\} \\
=\{x \mid 10 \leqslant x<100, x \in \mathbb{N}\} .
\end{array}
$$

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