Question 43
Let $A$ be a set with $n \ge 2$ elements and let $f_1, \dots, f_k : A \to \mathbb{R}$ be functions. Prove that there exist functions $g_1, \dots, g_{n-1} : A \to \mathbb{R}$ such that
$$
\max_{1 \le i \le k} |f_i(x) - f_i(y)| = \max_{1 \le i \le n-1} |g_i(x) - g_i(y)|
$$
for any $x, y \in A$.
$$
\max_{1 \le i \le k} |f_i(x) - f_i(y)| = \max_{1 \le i \le n-1} |g_i(x) - g_i(y)|
$$
for any $x, y \in A$.