Bounded filtered complex spectral sequence abuts to filtered homology

Fix n,p and write $Z=\ker d_n$, $D=\mathrm{im}{d}_{n+1}$. Choose R≥1 so that $F_{p-r}{C}_{n-1}=0$ and $F_{p+r-1}{C}_{n+1}=C_{n+1}$ for every r≥R; [F1] supplies bounds in precisely these two degrees. Then $A_{p,n}^r=F_p{C}_n\cap Z$ and $A_{p-1,n}^{r-1}=F_{p-1}{C}_n\cap Z$. Also $A_{p+r-1,n+1}^{r-1}=d_{n+1}^{-1}(F_p{C}_n)$, whose image is $F_p{C}_n\cap D$ by the epic-pullback calculation [F7].

The page quotient is therefore stationary with value $\frac{F_p{C}_n\cap Z}{(F_{p-1}{C}_n\cap Z)+(F_p{C}_n\cap D)}\mathrm{.}$ Its projected numerator and boundary subobjects are stationary too, giving exactly $E^{\mathrm{\infty }}$ under [F4]. By [F2] this quotient is canonically ${\mathrm{gr}}_p{H}_n(C)$. Separately the initial page on diagonal n vanishes outside $a_n<p\le b_n$, so [F5] supplies vanishing of both incident differentials and stabilization in the specified sense.

For $p\le a_n$, the cycle subobject $F_p{C}_n\cap Z$ is zero, so the image filtration on $H_n(C)$ is zero. For $p\ge b_n$ it is Z, whose quotient map surjects onto $Z\mathrm{/}D=H_n(C)$; hence $F_p{H}_n(C)=H_n(C)$. Thus the filtration is finite. A family containing 0 and $H_n$ has meet 0 and join $H_n$ by the subobject order, proving separation and exhaustiveness without infinite exactness.

A filtered chain map preserves the cycle and boundary subobjects in steps 1.1–2.1, and all quotient comparisons commute by [F7]. The page transitions [F3] were built from the same cycle inclusions; their stationary identifications agree with these comparisons. Thus the displayed isomorphisms form natural abutment data in the sense of [F6]. Each argument fixed n before using its finite bounds, so no common bounds over all degrees were needed.