Let $B$ be a unital Banach algebra and $A$ be a Banach subalgebra of $B$ such that $1_B \in A.$ Let $a_n \in GL(A)$ and $a = \lim\limits a_n.$ If $a \in GL(B),$ then show that $a \in GL(A).$
This is one of the exercises which has been left as an exercise in the lecture note I am following which I am unable to figure out properly. A small hint would be very much appreciated at this stage.
Thanks for your time.