Just trying to understand the text below.
Question 1: What exactly is the argument below (1)? "This is enough because applying (1) to $v=x-y$ for $x,y\in\mathbb{R}^n$, we get $f(x,y)=f(x)-f(y)$, and by a union bound over $\pmatrix{n\cr2}$ pairs $x\ne y\in X$ the theorem follows" where are $x,y$ come from? How the union bound is applied? If $X$ is a subspace of $l^2$, then there should be many more points than $\pmatrix{n\cr2}$.
Question 2: How is 3 chosen in the second to last line? Is this important?
Here is the main theorem this is trying to prove.
And here's the link to the text http://www.wisdom.weizmann.ac.il/~robi/teaching/2009b-SeminarGeometryAlgs/scribe5.pdf