I am trying to prove Theorem 3 from this paper regarding invariant subspaces. The paper first proves the following (Theorem 2):
\begin{align} ||\hat{F}^T(t)F_\perp (t-b) ||_2 \leq \frac{||K-\hat{K}||_2}{b} \\ ||F^T(t+b) \hat{F}_\perp (t) ||_2 \leq \frac{||K-\hat{K}||_2}{b} \end{align}
for any $b>0$, where $|| \cdot ||_2 = \sigma_{max}(\cdot)$ denotes the L2 norm and $K$ and $\hat{K}$ are symmetric psd with singular value decompositions given by \begin{align} K = F D F^T \\ \hat{K} = \hat{F} \hat{D} \hat{F}^T \end{align}
and $F(t)$ denotes the column space of $F$ containing eigenvectors with eigenvalue $\geq t$ (and $F_\perp(t)$ is the orthogonal complement of $F(t)$, containing eigenvectors with eigenvalue $< t$).
Then they state that this leads directly to the following (Theorem 3):
$$ || \hat{F}^T(\lambda_\ell)\hat{v} - R\cdot F^T(\lambda_\ell)v|| \leq \frac{||v||}{c} $$
where $v$ is any column of $K$, $\hat{v}$ is the corresponding column of $\hat{K}$, and $R$ is some orthonormal matrix. They say to simply instantiate Theorem 2 with $b=2 c ||K - \hat{K}||_2$ and $t = \lambda_\ell$, however I cannot get anywhere close to proving the desired result.
Part of the difficulty is that Theorem 2 refers to products of projectors while Theorem 3 describes differences of vectors projected onto the corresponding subspaces. I have tried to re-derive Theorem 2 using vectors $v$ and $\hat{v}$ instead of $K$ and $\hat{K}$ but I couldn't get both $\hat{F}(t)$ and $F(t)$ to appear transforming $v$ and $\hat{v}$ seperately.
I have also tried rewriting terms in Theorem 2 as differences of appropriately padded matrices, e.g. $F_\perp (t) = F - F(t)$ , but again neither inequality contains both $\hat{F}(t)$ and $F(t)$ so I couldn't move forward.
I'd appreciate any suggestions on how to approach this since I'm unfamiliar with this kind of analysis.