You can prove this using Zorn's lemma on the pairs (D,h) where D is a subspace of B and h is a partial section.
And yes, you do need Zorn's lemma: without it, there may exist vector spaces none of whose nontrivial subspaces has a complement (Herrlich, Axiom of Choice, LNM 1876, Disaster 4.43).
A very readable introduction to spectral sequences is Chapter III of
In particular you can find details about both Mayer-Vietoris and Leray Spectral sequences.
Another good reference:
I just would like to remark that many important spectral sequences are particular cases of the Grothendieck spectral sequence for derived functor of the composition of two functor. For instance the Leray spectral sequence and the exact sequence of low degrees.
Let
$$E_{2}^{h,k} \Longrightarrow H^{n}(A)$$
be a spectral sequence whose terms are non trivial only for $h,k \geq 0$. Then we have
$$0\mapsto E^{1,0}_{2}\rightarrow H^{1}(A) \rightarrow E^{0,1}_{2} \rightarrow E^{2,0}_{2} \rightarrow H^{2}(A).$$
Let $\mathcal{F}:\mathcal{C}_{1}\rightarrow \mathcal{C}_{2}$ and $\mathcal{G}:\mathcal{C}_{2}\rightarrow \mathcal{C}_{3}$ be two additive covariant functors between abelian categories. Suppose that $\mathcal{G}$ is left exact and that $\mathcal{F}$ takes injective objects of $\mathcal{C}_{1}$ in $\mathcal{G}$-acyclic objects of $\mathcal{C}_{2}$. Then there exists a spectral sequence (Grothendieck spectral sequence) for any object $A$ of $\mathcal{C}_{1}$
$$E_{2}^{h,k} = (R^{h}\mathcal{G}\circ R^{k}\mathcal{F})(A) \Longrightarrow R^{h+k}(\mathcal{G}\circ \mathcal{F})(A).$$
The corresponding exact sequence of low degrees is the following
$$0\mapsto R^{1}\mathcal{G}(\mathcal{F}(A)) \rightarrow R^{1}(\mathcal{G}\mathcal{F}(A)) \rightarrow \mathcal{G}(R^{1}\mathcal{F}(A)) \rightarrow R^{2}\mathcal{G}(\mathcal{F}(A)) \rightarrow R^{2}(\mathcal{G}\mathcal{F})(A).$$
As a special case of the Grothendieck spectral sequence we get the Leray spectral sequence. Let $f:X\rightarrow Y$ be a continuous map between topological spaces. We take $\mathcal{C}_{1} = \mathfrak{Ab}(X)$ and $\mathcal{C}_{2} = \mathfrak{Ab}(Y)$ to be the categories of sheaves of abelian groups over $X$ and $Y$ respectively. Then we take $\mathcal{F}$ to be the direct image functor $f_{*}:\mathfrak{Ab}(X)\rightarrow \mathfrak{Ab}(Y)$ and $\mathcal{G} = \Gamma_{Y}:\mathfrak{Ab}(Y) \rightarrow \mathfrak{Ab}$ to be the global section functor, where $\mathfrak{Ab}$ is the category of abelian groups. Note that
$$\Gamma_{Y}\circ f_{*} = \Gamma_{X}:\mathfrak{Ab}(X) \rightarrow \mathfrak{Ab}$$
is the global section functor on $X$. By Grothendieck's spectral sequence we know that $(R^{h}\Gamma_{Y}\circ R^{k}f_{*})(\mathcal{E}) \Longrightarrow R^{h+k}(\Gamma_{Y}\circ f_{*})(\mathcal{E}) = R^{h+k}\Gamma_{X}(\mathcal{E})$ for any $\mathcal{E} \in \mathfrak{Ab}(X)$, that is
$$H^{h}(Y,R^{k}f_{*}\mathcal{E}) \Longrightarrow H^{h+k}(X,\mathcal{E}).$$
The exact sequence of low degrees looks like
$$0\mapsto H^{1}(Y,f_{*}\mathcal{E})\rightarrow H^{1}(X,\mathcal{E})\rightarrow H^{0}(Y,R^{1}f_{*}\mathcal{E}) \rightarrow H^{2}(Y,f_{*}\mathcal{E})\rightarrow H^{2}(X,\mathcal{E}).$$
Finally we can work out the local to global spectral sequence of Ext functors. Let $\mathcal{E} \in \mathfrak{Coh}(X)$ be a coherent sheaf on a scheme $X$. Consider the functor
$$\mathcal{H}om(\mathcal{E},-):\mathfrak{Coh}(X) \rightarrow \mathfrak{Coh}(X), \: \mathcal{Q}\mapsto \mathcal{H}om(\mathcal{E},\mathcal{Q}),$$
and the global section functor
$$\Gamma_{X}:\mathfrak{Coh}(X) \rightarrow \mathfrak{Ab}, \: \mathcal{Q}\mapsto \Gamma_{X}(\mathcal{Q}).$$
Note that $\Gamma_{X}\circ \mathcal{H}om(\mathcal{E},-) = Hom(\mathcal{E},-)$. By Grothendieck spectral sequence we have $(R^{h}\Gamma_{X}\circ R^{k}\mathcal{H}om(\mathcal{E},-))(\mathcal{Q}) \Longrightarrow R^{h+k}(Hom(\mathcal{E},-)(\mathcal{Q})$ for any $\mathcal{Q}\in \mathfrak{Coh}(X)$, that is
$$H^{h}(X,\mathcal{E}xt^{k}(\mathcal{E},\mathcal{Q})) \Longrightarrow Ext^{h+k}(\mathcal{E},\mathcal{Q}).$$
The corresponding sequence of low degrees is
$$0\mapsto H^{1}(X,\mathcal{H}om(\mathcal{E},\mathcal{Q})) \rightarrow Ext^{1}(\mathcal{E},\mathcal{Q}) \rightarrow H^{0}(X,\mathcal{E}xt^{1}(\mathcal{E},\mathcal{Q}))\rightarrow H^{2}(X,\mathcal{H}om(\mathcal{E},\mathcal{Q})) \rightarrow Ext^{2}(\mathcal{E},\mathcal{Q}).$$
Best Answer
There is one obvious sequence that underlies all vector analysis and a lot that builds up on it, no matter if its applied analysis, PDE, physics or the original foundations of algebraic topology. Yet it is rarely written out, as the people in the applied fields prefer to split it into its constituent statements and the people in pure mathematics are inclined to immediately write down some generalization instead. What I am talking about is of course the relationship between the classic differential operators on 3D vector fields:
$$0 \to \mathbb R\to C^\infty(\mathbb{R}^3;\mathbb{R}) \stackrel{\operatorname{grad}}{\to} C^\infty(\mathbb{R}^3;\mathbb{R}^3) \stackrel{\operatorname{curl}}{\to} C^\infty(\mathbb{R}^3;\mathbb{R}^3) \stackrel{\operatorname{div}}{\to} C^\infty(\mathbb{R}^3;\mathbb{R}) \to 0 $$