In his paper Axial Anomalies and Index Theorems on Open Spaces, Callias provides a wonderful index theorem $$\mathrm{index}(L)=\lim_{z\to0} \mathrm{Tr}B_z\quad\text{where} \quad B_z=\frac{z}{L^\dagger L+z}-\frac{z}{LL^\dagger+z}$$ for $L:D\to H$ a closed Fredholm operator on a dense subset $D$ of the Hilbert space $H$ and $z\in\mathbb C$.
The proof assumes that $B_z$ is trace-class on $H$, and that $Tr |B_z|$ is bounded for $z$ in some domain $C\in\mathbb C$ with $0\in \overline C$. It considers the expression $$ \widetilde B_z=\frac{z}{L^\dagger L+z}-Pr_{\mathrm{Ker}(L^\dagger L)}-\frac{z}{LL^\dagger+z}+Pr_{\mathrm{Ker}(LL^\dagger)}$$ where $Pr_S$ denote the projection operator on the linear subspace $S$ of $H$. The author then asserts without comment that $\lim_{z\to0} \widetilde B_z=0$ strongly, from which the index theorem follows in a straightforward manner.
How would one go about proving that $ \widetilde B_z$ indeed converges strongly to the zero operator as $z$ approaches zero?