I'm learning homological algebra using Weibel's book. And I have trouble in exercise 5.4.4:
Let $f : B \to C$ be a map of filtered chain complexes. For each $r \geq 0$, define a filtration on the mapping cone $cone(f)$ by $$F_p cone(f)_n = F_{p-r} B_{n-1} \oplus F_p C_n.$$ Show that $E^r_p (cone f)$ is the mapping cone of $f^r : E^r_p (B) \to E^r_p (C)$.
I'm not used to deal with mapping cone and spectral sequence, and I'm curious about the approach to this problem.
At first, I tried to represent $E^r_p (conef)$ explicitly by the construction process of $E^r_p$, but the indexing and calculation were so complicated that I doubted if it was the right way.
Is there a hint about this exercise?