Alternative proof of the a priori $\tan\Theta$ theorem

Let $A$ be a self-adjoint operator in a separable Hilbert space. We assume that the spectrum of $A$ consists of two isolated components $\sigma_0$ and $\sigma_1$ and the set $\sigma_0$ is in a finite gap of the set $\sigma_1$. It is known that if $V$ is a bounded additive self-adjoint perturbation of $A$ that is off-diagonal with respect to the partition $\operatorname{spec}(A)=\sigma_0\cup\sigma_1$, then for $\|V\|<\sqrt{2}d$, where $d= \operatorname{dist}(\sigma_0,\sigma_1)$, the spectrum of the perturbed operator $L=A+V$ consists of two isolated parts $\omega_0$ and $\omega_1$, which appear as perturbations of the respective spectral sets $\sigma_0$ and $\sigma_1$. Furthermore, we have the sharp upper bound $\|\mathsf{E}_A(\sigma_0)- \mathsf{E}_L(\omega_0)\|\le\sin\bigl(\arctan(\|V\|/d)\bigr)$ on the difference of the spectral projections $\mathsf{E}_A(\sigma_0)$ and $\mathsf{E}_L(\omega_0)$ corresponding to the spectral sets $\sigma_0$ and $\omega_0$ of the operators $A$ and $L$. We give a new proof of this bound in the case where $\|V\|<d$.