**Edit:**


Let $X$ be a strict [LB-space](https://en.wikipedia.org/wiki/LB-space) described by $\lim X\_n$ and suppose that $\{x\_n\}\_{n \in \mathbb{N}}$ converges in $X$. I'm looking for a reference showing that $x\_n$ must converge in some $X\_N$. 



Convergence in LB-spaces
The result (even for LF-spaces) is due to J. Dieudonné and L. Schwartz
*La dualité dans les espaces (F) et (LF)*, Annales de l’institut Fourier, tome 1 (1949), p. 61-101, propositions 2 and 4.


(Proposition 2 says that the inductive limit topology induces on the ,,steps'' their original topologies, and proposition 4 says that bounded subsets of the inductive Limit are contained in steps.)


