Let $f_1(x), \ldots, f_n(x)$ be scalar functions. Suppose that every set of sample points $x_1, \ldots, x_m \in \mathbb{R}$, for all finite $m \geq 1$, leads to linearly dependent sample vectors $\mathbf{f}_1, \ldots, \mathbf{f}_n \in \mathbb{R}^m$. Prove that $f_1(x), \ldots, f_n(x)$ are linearly dependent functions.
Hint: Given sample points $x_1, \ldots, x_m$, let $V_{x_1, \ldots, x_m} \subset \mathbb{R}^n$ be the subspace consisting of all vectors $\mathbf{c}=\left(c_1, c_2, \ldots, c_n\right)^T$ such that $c_1 \mathbf{f}_1+\cdots+c_n \mathbf{f}_n=\mathbf{0}$. First, show that one can select sample points $x_1, x_2, x_3, \ldots$ such that $\mathbb{R}^n \supsetneq V_{x_1} \supsetneq V_{x_1, x_2} \supsetneq \cdots$. Then, apply Exercise 2.4 .25 to conclude that $V_{x_1, \ldots, x_n}=\{0\}$.