Proposition 4

Let A be a unital Banach algebra. Let $x$ be an element of A which is not left invertible. If there exists a sequence $(x_n)$ of left invertible elements of A which converges to $x$, then $x$ is a topological right zero divisor.

Let $y_n$ be a left inverse of $x_n$. By Prop. 3 (ii), $\|y_n\|$ tends to $+\infty$. Let $z_n=\|y_n\|^{-1}y_n$. We have $\|z_n\|= 1$, and $z_nx_n=\|y_n\|^{-1}$ tends to 0, hence $z_nx=z_nx_n+z_n(x-x_n)$ tends to 0. One then concludes with the aid of the remark following Definition 4.