If $A$ and $B$ are continuous linear operators from $F$ into $E$, $|A-B|<1 $, and $A$ is invertible, then $B$ is invertible

Question

Let $E$ be a Banach space and $F$ a normed space. Assume that $A$ and $B$ are continuous linear operators from $F$ into $E$, $|A-B|<1 $, and $A$ is invertible. Then I want to show that $B$ is invertible.

I know that if $T$ is any continuous linear operator from $F$ into $E$ with $|T|<1$, then $1-T$ is invertible and its inverse is $\sum_{n=0}^{\infty}T^{n}(x)$. I think I need to use this fact to show that $B$ is invertible. But I couldn't see how to do it. I need some hint.

Answer 1

Let me advocate for "Not true":
Assume $A$ is invertible and has norm smaller than one. Then consider $B=0$.