Let $\mathcal{H}$ be a Hilbert space. If $\left(A_n\right)$ is a norm-convergent sequence of bounded, bounded-below, but non-surjective operators such that the limit operator $A$ is also bounded and bounded below. $\text{Ran}(A_n)$ has fixed dimension/condimension. Is it possible for $A$ to be surjective? Or must $A$ preserve the dimension/codimension? How about on Banach spaces?
Thanks in advance.