1. Given non-empty sets
$$
\begin{array}{l}
A=\{x \mid m+1 \leqslant x \leqslant 2 m-1\}, \\
B=\left\{x \mid x^{2}-2 x-15 \leqslant 0\right\},
\end{array}
$$

and $A \subseteq B$.
Then the range of real number $m$ is
-, 1. $[2,3]$.
It is easy to know that $B=\{x \mid-3 \leqslant x \leqslant 5\}$.
Since $A \subseteq B$ and $A \neq \varnothing$, we have
$$
\begin{array}{l}
-3 \leqslant m+1 \leqslant 2 m-1 \leqslant 5 \\
\Rightarrow 2 \leqslant m \leqslant 3 .
\end{array}
$$