I have a possibly simple question regarding estimating bounds on spectral projection subspace.

Let $H_n$ be a sequence of bounded self-adjoint operators on $\ell^2(\mathbb{Z}^2)$ converging in the strong operator topology to $H_\infty$. I assume that $\dim \big(\text{Im}\chi_{(a-\epsilon,a+\epsilon)}(H_n) \big)=m_n\in \mathbb{N}$ for all $n$. It seems that this should imply that $\dim \Big(\text{Im}\big(\chi_{[a-\epsilon,a+\epsilon]}(H_\infty) \big) \Big)\geq \liminf m_n$. I would also like to say that if $m_n\to \infty$, then $\sigma_{ess}(H_\infty)\cap [a-\epsilon,a+\epsilon]\neq \emptyset$.

Is this result or something of the sort a known result or something that anyone thinks should hold? I want to check whether I can deduce a spectral type statement about the limit in the strong operator topology.