If $A$ is positive definite then any principal submatrix of $A$ is positive definite

The $j_i$'s represent the coordinates removed from $A$ to obtain the principal submatrix $B$. So we want to show $y^{\intercal}By > 0$ for all non-zero $y \in \mathbb{R}^{n-s}$. The argument shows that any such $y$ can be exported to some $x \in \mathbb{R}^n$ (by choosing the missing coordinates as zero) such that $y^{\intercal}By = x^{\intercal}Ax$. Then the result follows.