The following question is from my exercise sheet in Banach Algebras and I am not able to solve 1 part.
Question: Let $A$ be a Banach Algebra with $1$.
(a) Let $a,b\in A$, then show that $1-ab$ is invertible iff $1-ba$ is invertible.
(b) Let $a,b$ be such that $\|a\|<1$ and $\|b\|<1$. Show that $1-ab $ and $1-ba$ are invertible in $A$.
Attempt: I have shown (a) using the following obeservation: if $1−ab$ has inverse $x$ , then $1−ba$ has inverse $1+bxa$.
But I am not able to prove (b) part.
Can you please tell me how should I approach this part?