A filtered complex produces an exact couple

Since $d$ preserves $F_{p-1}C\subseteq F_{p}C$, it induces a unique differential on each quotient $G_{p}C=F_{p}C\mathrm{/}{F}_{p-1}C$. Its square is zero after precomposition with the epic quotient, since $d^2=0$. The inclusion and quotient therefore form chain maps. In each degree the inclusion is a kernel of its cokernel, so $0\to F_{p-1}C\to F_{p}C\to G_{p}C\to 0$ is a short exact sequence of complexes.

With $n=p+q$, the homology sequence contains $H_n(F_{p-1}C)\to H_n(F_{p}C)\to H_n(G_{p}C)\to H_{n-1}(F_{p-1}C)\to H_{n-1}(F_{p}C)$. In the proposed notation these arrows are $D_{p-1,q+1}^1\stackrel{i}{\to }{D}_{p,q}^1\stackrel{j}{\to }{E}_{p,q}^1\stackrel{k}{\to }{D}_{p-1,q}^1\stackrel{i}{\to }{D}_{p,q-1}^1$. This calculates the degrees of all three maps, including the $q$ coordinate of the connector.

Exactness of this sequence gives $\mathrm{im}i=\ker j$ in $D_{p,q}^1$, $\mathrm{im}j=\ker k$ in $E_{p,q}^1$, and $\mathrm{im}k=\ker i$ in $D_{p-1,q}^1$. Letting $(p,q)$ range over all integers gives every vertex required by the initial exact-couple definition. The argument applies when adjacent filtration pieces coincide or vanish; their zero quotient causes no exception. It treats the families componentwise and never takes an infinite sum of exact sequences, so no infinite exactness or choice hypothesis is used.