suppose $A$ is a C* algebra and I consider a sequence of non invertible elements $a_n$ which is bounded below and bounded above in norm. I'm wanting to show that it is not possible to pick a sequence $c_n$ of elements in $A$ whose norms converge to zero such that $a_n+c_n$ is invertible for all $n$. I'm not being able to show this so i'm starting to think it might be wrong. Does anyone have a counterexample? Or if it is true does anyone know how to show this?
2026-03-30 12:19:31.1774873171
spectrum of a sequence
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