Let $A=\int_{\sigma(A)} \lambda dP(\lambda)$ be an unbounded self-adjoint operator on a Hilbert space $H$ with dense domain $D\subset H$.
Let $A_n=\int_{\sigma(A)\cap[-n,n]} \lambda dP(\lambda)$ (see motivation), which then defines a sequence of bounded (by $n$, from spectral radius) self-adjoint operators $\{A_n\}$. I want to show strong convergence $A_n \rightarrow A$, e.g. $||A_n\psi-A\psi||$ converges for all $\psi \in D$. Since $A$ is unbounded and $\psi$ has infinite support in general, I am struggling to come up with a convergence argument, not least because I think the operator $A$ (smeared field operator in QFT, but could be position operator in QM also) should have a spectrum that is $\mathbb{R}$, so $A_n-A$ has a spectrum $\sim (-\infty,-n)\cap(n,\infty)$, and I am not hugely familiar with these operators outside of the usual countable dimension assumptions in QM. If A were bounded, I would make an argument based on existence of an $n > $spectral radius, but of course that does not help here. Any hints would be much appreciated.
Just use $$ \|(A-A_n)\psi\|^2=\int_{\sigma(A)\cap(-\infty, -n)} \lambda^2\;\mathrm{d}\|P(\lambda)\psi\|^2+\int_{\sigma(A)\cap(n ,\infty)} \lambda^2\;\mathrm{d}\|P(\lambda)\psi\|^2, \quad \psi \in D$$ and apply monotne convergence theorem (the integrals above are just standard integrals in $\mathbb{R}$).