I'm trying to understand the proof of a theorem taken from a textbook.
Theorem: If $S_1,S_2$ are disjoint non-empty convex subsets of a real vector space $X$ (which may be infinite dimensional) then there is some non-zero linear functional $f:X\to \mathbb{R}$ (which may be unbounded) such that, for some constant $c$, $f(s_1) \leq c \leq f(s_2)$ for all $s_1 \in S_1$, $s_2 \in S_2$.
My question #1: The proof starts by assuming wlog that $S_1$ is not contained in a hyperplane. Why is this wlog? I know that if $S_1$ is contained in a hyperplane, there is a linear functional whose kernel is this hyperplane. But that doesn't mean $f(s_2) \geq 0$ for that particular $f$? Note that $S_1$ may be a proper subset of a hyperplane.
My question #2: The author goes on to say that, assuming $S_1$ is not contained in a hyperplane, we can translate $S_1,S_2$ so that for all $x \in X$, there is some $\epsilon(x)>0$ such that the line segment $[-\epsilon(x) x, \epsilon(x) x]$ is contained in $S_1$. Why?