Let $\theta \in [0,2\pi]$, and let $\theta_j$ a sequence such that $\forall j \; \theta_j \in [0,2\pi]$, $\theta_j \to \theta$ with the property
$$ |\theta_j - \theta| \geq |\theta_{j+1} - \theta|. $$
Let $R_{\psi}$ the rotation matrix, counter clockwise, respect to $\psi$. By drawing something on paper I can see that for all $x \in \mathbb{R}^2$ I have
$$ \lVert \left(R_{\theta_j - \theta} - I\right)x \rVert_2 \geq \lVert \left(R_{\theta_{j+1} - \theta} - I\right)x \rVert_2 $$
But how can I prove this formally?
My attempt were probably too convoluted, mainly based on matrix norms to derive bounds, but I haven't actually got anything yet. Even a clue would be highly appreciated.