# Continuously bounding a matrix valued function by another matrix valued function

Notation: The symbol $\mathbb{S}^{n \times n}$ denotes the set of symmetric matrices of order $n \times n$ and $A \succ 0$ denotes that $A$ is a positve definite matrix.

Let $M:S \rightarrow \mathbb{S}^{n\times n}$ and $N:S \rightarrow \mathbb{S}^{n\times n}$ be continuous symmetric matrix valued functions defined on a compact set $S \subset \mathbb{R}$.

If for each $s \in S$ there is an $\alpha \in \mathbb{R}$ (possibly dependent on s) such that $M(s) \prec \alpha N(s)$ then there exists a continuous function $\bar{\alpha}:S \rightarrow \mathbb{R}$ such that $M(s) \prec \bar{\alpha}(s) N(s) \, ?$

I have tried to use Michael selection theorem (which maybe is overkill) on the multivalued function $S \rightrightarrows \mathbb{R}$ given by $$s \mapsto \{\alpha \in \mathbb{R} \mid M(s) \prec \alpha N(s) \}$$ but I'm having difficulty in proving that this function is lower hemicontinuous...