Is closure of a proper ideal in a unital Banach algebra also proper?

Question

Let $A$ be a unital Banach algebra. Let $J$ be a proper ideal of $A.$ Can it be concluded that $\overline {J}$ is also a proper ideal of $A\ $?

I know that closure of an ideal is also an ideal but I don't know whether the closure of a proper ideal is proper or not. Can anybody please help me in this regard?

Thanks for your time.

Answer 1

If $\overline J = A$ then you can get hold of a sequence $(x_n)_{n \geq 1}$ in $J$ converging to $1.$ So there exists some $N \in \mathbb N$ such that for all $n \geq N$ we have $\|1 - x_n\| \lt 1.$ But then $x_n$ would be invertible for all $n \geq N,$ contradicting the properness of $J.$