Let $(E_\alpha,\tau_\alpha,g_\alpha)$ be a family of bornological (locally convex) topological vector spaces $(E_\alpha,\tau_\alpha)$, where a LCTVS $E$ is said to be *bornological* if every circled, bounded subset $A\subset E$ that absorbs every bounded set is a neighborhood of zero (i.e. has nonempty interior), and where $g_\alpha:E_\alpha\to E$ induces an inductive LCTVS structure on $(E,\tau)$.

Then in Shaefer's book *Topological Vector Spaces*, section II.8.2, we are told that $E$ must be bornological. However, I am slightly confused about a detail of the proof.

Let $A$ be a convex, circled subset of $E$ absorbing all bounded sets. If $B_\alpha$ is bounded in $(E_\alpha,\tau_\alpha)$ then $g_\alpha(B_\alpha)$ is bounded in $E$, then $A$ absorbs $g_\alpha(B_\alpha)$, whence $g_\alpha^{-1}(A)$ absorbs $B_\alpha$, and so $g_\alpha^{-1}(A)$ contains a neighborhood of $0$ in $E_\alpha$. Since this holds for all $\alpha$, then $A$ contains a neighborhood of $0$ in $E$.

However, I'm very confused about the last step — the inductive limit property does not directly give us the existence of opens in $E$, it merely places a constraint on them. So how do we figure out that $A$ contains a neighborhood of $0$ in $E$?