Exercise 13, § 4

With the hypotheses of II, § 6, show that E is complete for topology $\mathscr{T}$ which is the inductive limit of the $\mathscr{T}_n$, if and only if, for each integer n and every Cauchy filter $\mathfrak{F}$ on $E_n$ for the topology induced by $\mathscr{T}$, there exists $p \geq n$ such that $\mathfrak{F}$ is convergent in $E_p$ for the topology $\mathscr{T}_p$.