Given a symmetric matrix A, of dimension n×n and an arbitrary unitary matrix V of dimension n×p. we define the trace ratio problem $$ \left\{\begin{array}{cc} \max _{V \in \mathbb{R}^{n \times p}} & \frac{\operatorname{Tr}\left[V^{T} A V\right]}{\operatorname{Tr}\left[V^{T} B V\right]} \\ V^{T} C V=I \end{array}\right. $$ where B and C are assumed to be symmetric and positive definite.
I want to show that it exists a maximum for this problem?
I know that is sufficient to prove that the function $f(V)=\frac{\operatorname{Tr}\left[V^{T} A V\right]}{\operatorname{Tr}\left[V^{T} B V\right]}$ is a continuous function over the closed set of matrices V satisfying $V^T CV $.
My question is how to prove that the set above is a closed set?