Let $L_1$, $L_2$ be two subspaces in $\mathbb{R}^n$ and $\dim(L_1) = \dim(L_2) = s<n$. Let $$0 \leq \theta_1 \leq \dotsb\leq \theta_{s} \leq \pi/2$$ be the principal angles between $L_1, L_2$. Let there be two unit vectors $u_1\in L_1$ and $u_2 \in L_2$ and the principal angle between $\operatorname{span}\{u_1\}$ and $\operatorname{span}\{u_2\}$ is $\alpha \in [0,\pi/2]$. Is there any known upper bound on $\alpha$ in terms of $\theta_{s}$?

Here is my try: Let $U_1, U_2 \in \mathbb{R}^{n \times s}$ be two orthogonal matrices such that their columns are orthonormal bases of $L_1$ and $L_2$ respectively. Now any unit vector in $L_j$ is essentially of the form $U_j \alpha_j$ where $\lVert \alpha_j\rVert_2 =1 $ for $j \in \{1,2\}$. Also, the singular values of $U_1^\top U_2$ are $\{\cos \theta_1, \dotsc, \cos \theta_s\}$. Now, I don't have any idea how to proceed any further. Any help will be appreciated.