Exercise 22, § 2

In $\mathbf{R}^2$ consider a set of $2m$ points of the form $(a_i, b'_i), (a_i, b''_i)$ where $b'_i \leq b''_i$ for $1 \leq i \leq m$. Let $n$ be an integer $< m - 2$. In order that there should exist a polynomial $P(x)$ of degree $\leq n$ such that $b'_i \leq P(a_i) \leq b''_i$ for $1 \leq i \leq m$, it is sufficient that, for every family $(i_k)_{1 \leq k \leq n+2}$ of $n + 2$ indices $i$, there exists a polynomial $Q(x)$ of degree $\leq n$ such that $b'_{i_k} \leq Q(a_{i_k}) \leq b''_{i_k}$ for every integer $k$ such that $1 \leq k \leq n + 2$. (Use exerc. 21, a).)