Bounded filtered-complex convergence theorem ($E_1^{p,q}=H^{p+q}(F^{p}C/F^{p+1}C)\Rightarrow H^{p+q}(C)$)

A finite filtration of a cochain complex produces a convergent spectral sequence. The induced abutment filtration is the image of $H^n(F^{p}C)\to H^n(C)$. Finiteness in each degree suffices; a bounded range of cochain degrees is not required.