I was going through this proof from Rotman's 'Introduction to homological algebra' (Pages 381-382) and I just can't seem to make sense of it, am not super well-versed in this so I don't know if it's too short and they're assuming that I know things I don't know or it's just simply going over my head. Here is the fragment verbatim:
Let $\mathcal{F}$ be a sheaf over a space $X$.
(i) If 0 \rightarrow \mathcal{F}' \xrightarrow{\iota} \mathcal{F} \xrightarrow{\varphi} \mathcal{F}'' \rightarrow 0 is an exact sequence of sheaves with \mathcal{F}' flabby, then 0 \rightarrow \Gamma(\mathcal{F}') \rightarrow \Gamma(\mathcal{F}) \rightarrow \Gamma(\mathcal{F}'') \rightarrow 0 is an exact sequence of abelian groups.
PROOF.
It suffices to prove that \varphi_X:\Gamma(\mathcal{F}) \rightarrow \Gamma(\mathcal{F}''), given by $\varphi_X:s \mapsto \varphi s$, is epic. Let s'' \in \mathcal{F}''(X) = \Gamma(\mathcal{F}''). Define
$\mathcal{X} = \{ (U,s):U \subseteq X$ is open, s\in \mathcal{F}(U),\varphi s=s''\mid U \} .
Partially order $\mathcal{X}$ by $(U,s) \preceq (U_1,s_1)$ if $U \subseteq U_1$ and $s_1 \mid U = s$. It is routine to see that chains in $\mathcal{X}$ have upper bounds, and so Zorn's Lemma provides a maximal element $(U_0,s_0)$. If $U_0=X$, then $s_0$ is a global section and $\varphi_X$ is epic. Otherwise, choose $x \in X$ with $x \notin U_0 $. Since \varphi:\mathcal{F} \rightarrow \mathcal{F}'' is an epic sheaf map, it is epic on stalks, and so there are an open $V \subseteq X$ with $V \ni x$ and a section $t \in \mathcal{F}(V)$ with \varphi t = s'' \mid V. Now s - t \in \mathcal{F}'(U \cap V) (we regard \iota: \mathcal{F}' \rightarrow \mathcal{F} as the inclusion), so that \mathcal{F}' flabby provides r \in \mathcal{F}'(X) extending $s - t$. Hence, $s = t + r \mid (U \cap V)$ in $\mathcal{F}(U \cap V)$. Therefore, these sections may be glued: there is $\tilde{s} \in \mathcal{F}(U \cup V)$ with $\tilde{s} \mid U = s$ and $\tilde{s} \mid V = t + r \mid (U \cap V)$. But \varphi(\tilde{s}) = s'', and this contradicts the maximality of $(U_0,s_0)$.
MY QUESTIONS ARE:
1 - Why does it say that 'it is ENOUGH to prove that $\varphi_X$ is epic'?
2 - Why does it say 'If $U_0 = X$, then $s_0$ is a global section and $\varphi_X$ is epic'? Why is $\varphi_X$ epic if $s_0$ is global?
3 - I got stuck there so the rest of the proof I'm basically in La La land too