Spectral sequences: equivalence of exact couples and classic (?) method

Question

By the 'classic' method I mean the construction of the spectral sequence associated to a filtration as found in Weibel's book p. 133-134. There is also the method of construction through exact couples due to Massey. I would like to find a proof that these two methods are equivalent. To elaborate, in the classic method we define certain objects $A^{r}_{p,p+q}$ and then objects $Z^{r}_{p,p+q}$ and $B^{r}_{p,p+q}$ from $A^{r}_{p,p+q}$ to eventually construct the spectral sequence $E^r$. With exact couples, we avoid this step but we can then define $Z^{r}_{p,p+q}$ as those cycles that 'never die' and $B^{r}_{p,p+q}$ as those elements that eventually bound. How do we show that these two definitions of $Z^{r}_{p,p+q}$ and $B^{r}_{p,p+q}$ are equivalent? I tried to do it but got bogged down in an inductive nightmare. A reference would be great.

Answer 1

You can find this in McClearly's "A user's guide to spectral sequences".

It is in Chapter 2 (page 42 in my edition). It's not as bad as you would think!