Noninvertible limits of invertibles have no one-sided inverse
= Noninvertible limits of invertibles have no one-sided inverse
Suppose $a_n\to a$ in a unital <Banach algebra>, with each $a_n$ invertible. If $ba=1$, then $ba_n\to1$, so $ba_n$ is invertible eventually by the <Neumann series>. Then $b=(ba_n)a_n^{-1}$ is invertible, and $a=b^{-1}$. The analogous argument with $a_nb\to1$ treats a <right inverse of an algebra element>. Thus a noninvertible such limit admits neither type of one-sided inverse.