I saw this question asked in some form somewhere, but couldn't quite prove the statement that I wanted to, so I am hoping that someone can help. Essentially I want to prove that a spectral sequence is a generalization of a long exact sequence.
Here's the formulation: Let $\mathcal{C}$ and $\mathcal{C'}$ be abelian categories and $F: \mathcal{C} \rightarrow \mathcal{C'}$ an additive left-exact functor. Assume that we know that for any chain complex $A^{\bullet}$ of objects in $\mathcal{C}$ there exist two spectral sequences $$ E_1^{p,q} = R^qF(A^p) \implies R^{p+q}F(A^{\bullet}) $$ and $$ E_2^{p,q} = R^pF(\mathcal{H}^q(A^{\bullet})) \implies R^{p+q}F(A^{\bullet}) $$ Let $0 \rightarrow B' \rightarrow B \rightarrow B'' \rightarrow 0$ be a short exact sequence in $\mathcal{C}$. Prove that from the existence statement above follows that we have a long exact sequence: $$ 0 \rightarrow R^0F(B') \rightarrow R^0F(B) \rightarrow R^0F(B'')\rightarrow R^1F(B') \rightarrow R^1F(B) \rightarrow \cdots $$