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?
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1Let $A=M_{2\times 2}(\Bbb C)$ and $a_n = \begin{pmatrix}1&0\0&0\end{pmatrix}$ and $c_n=\begin{pmatrix}0&0\0&\frac1n\end{pmatrix}$. The norms of the inverses of course go to infinity. – s.harp Dec 22 '19 at 14:36