So I'm working through Lay's Linear Algebra and its Applications, and I have just reached Chapter 8, the Geometry of Vector Spaces. The current section is Affine Combinations. Thus far, an affine set has been described as one where $(1-t){\bf{p}}+t{\bf{q}}\in S$ for ${\bf{p,q}}\in S$ and $t\in\mathbb{R}$.
Then comes along 'Theorem 2:'
What I gather from this is that 'S is affine if S is affine.' This is remarkably circular to me -- essentially, I don't understand this theorem.
Any help is appreciated!
In my reading, the theorem is about being closed under binary affine combinations implies being closed under arbitrary finite affine combinations (i.e. $x_i\in S\implies \sum_i\alpha_ix_i\in S$ whenever $\sum_i\alpha_i=1$).
And, for the proof, induction should be used.