An exact couple generates a spectral sequence

Applying the derived-couple theorem to any page-$r$ couple produces a page-$(r+1)$ couple whose $E$ object is exactly the homology of the preceding $jk$ differential. The associated differentials square to zero and their degrees are $(-r,r-1)$. Starting with the given initial couple and repeating this construction for each positive integer therefore supplies the objects, differentials and homology identifications required for a spectral sequence. The initial page is $E^1=E$.

The kernels of successive $i$ powers increase and their images decrease. Since $kj=0$, $B^r\subseteq N^s$ for every $r,s\ge 1$. Thus $B^r\subseteq B^{r+1}\subseteq N^{r+1}\subseteq N^r$ and each stated quotient exists. For $r=1$, $N^1=E$ and $B^1=j(\ker 1)=0$.

For $e$ through $N_{p,q}^r$, take a local $x$ at $D_{p-r,q+r-1}$ with $ke=i^{r-1}x$. The arrow $jx$ lies in every $N^s$ since $kjx=0$. Two such lifts differ by $\ker i^{r-1}$ and hence give the same class modulo $B^r$ in the target. Replacing $e$ by a local $B^r$ representative $jz$ changes $ke$ by zero. Consequently the rule $[e]\mapsto [jx]$ defines a unique map on $N^r\mathrm{/}{B}^r$, by quotient descent. Its degree is $(-r,r-1)$ and its square is zero: for the representative $jx$ its $k$ image is zero, so the next lift may be taken to be zero. At $r=1$ this rule is the original $jk$.

The kernel of this map is represented by exactly $N^{r+1}$. Indeed a zero image means locally $jx=jz$ with $i^{r-1}z=0$. Then $x-z\in \ker j=\mathrm{im}i$, so locally $x-z=iy$ and $ke=i^{r-1}x=i^{r}y$. Conversely if $ke=i^{r}y$, one may take $x=iy$ and then $jx=0$. Both containments descend. The incoming image is exactly $B^{r+1}\mathrm{/}{B}^r$: every output $jx$ has $i^{r}x=i(ke)=0$, and conversely if $i^{r}x=0$, then $i^{r-1}x\in \ker i=\mathrm{im}k$ has a local lift $e$ with $ke=i^{r-1}x$. This $e$ belongs to the appropriate $N^r$ and its image is $jx$. Thus the homology of the quotient at page $r$ is canonically $N^{r+1}\mathrm{/}{B}^{r+1}$.

To match these quotient pages with repeated derived couples in step 1.1, note that at the $r$th couple the $D$ object is $\mathrm{im}{i}^{r-1}$ inside the original $D$. Its $k$ map is induced by the original $k$ on $N^r$, and its $j$ map sends $i^{r-1}x$ to $[jx]$. These assertions hold initially. On deriving once, the image of the restricted $i$ is $\mathrm{im}{i}^r$; the new $k$ is induced by the same original $k$ on the new cycles in step 3.1; and taking one more $i$-preimage changes $i^{r}y$ into $i^{r-1}y$, so the new $j$ sends $i^{r}y$ to $[jy]$. Step 3.1 identifies the new homology quotient and its transition by the inclusion of its numerator. This proves the asserted compatibility at every stage of the iteration.

Hence the stated subquotients and differentials describe precisely the spectral sequence of the exact couple, with specified canonical transition isomorphisms. The zero couple gives zero quotients at all pages. The use of finite composites at each fixed $r$, and canonical kernels, images and quotients, requires neither infinite sums nor AC. No target filtration or abutment has been constructed or inferred.