why every absolutely convex absorbing and closed set is a zero neighborhood?

Question

I need to prove that
"If $E=\lim_{n} E_n $, the inductive limit of Banach spaces $E_n$ and $A$ is absolutely convex, absorbing and closed set in $E$, then $id_n^{-1}(A)$ is a zero neighborhood in $E_n$ for all $n$. Here, $id_n^{-1}$ identity map from $E$ to $E_n$ which is continuous linear map".

Answer 1

In a locally convex inductive limit $E=\lim E_n$ (with linking maps $i_n:E_n\to E$) an absolutely convex set $A$ is a $0$-neighbourhood if and only if all preimages $i_n^{-1}(A)$ are $0$-neighbourhoods in $E_n$.

If $E_n$ are Banach and $A$ is closed and absorbing than also $i_n^{-1}(A)$ is closed (continuity of $i_n$) and absorbing and then Baire's theorem (or the fact that Banach spaces are barrelled) implies that $i_n^{-1}(A)$ is a $0$-neighbourhood in $E_n$.

This argument neither uses the injectivity of $i_n$ nor that you have a countable inductive limit.