Non-monotonic functions on ordered sets

Question

I'm trying to prove that if $~~(A,<_A)~~$ and $~~(B,<_B)~~$ are linearly ordered sets and $~~f: A \rightarrow B~~$ is non-monotonic function than there exist points $~~a,b,c\in A~~$ such that $~~ a <_A b <_A c~~$ and ($f(b)<_B\min\{f(a),f(c)\}~~$ or $~~\max\{f(a),f(c)\}<_Bf(b)$)

But unfortunately nothing better than brute-force enumeration of possibilities comes to my mind. So i'll be grateful for any tips or solutions for this question. Thanks in advance.

Answer 1

We can try by contradicition.

The negation of the statement to be proved is :

for all points $a,b,c \in A \quad [ \lnot (a < b < c) \lor \lnot (f(b) < \min \{ f(a),f(c) \} \lor \max \{ f(a),f(c) \} < f(b)) ]$

which is :

for all points $a,b,c \in A \quad [ (a < b < c) \rightarrow (f(b) \ge \min \{ f(a),f(c) \} \land \max \{ f(a),f(c) \} \ge f(b)) ]$

which is :

for all points $a,b,c \in A \quad [ (a < b < c) \rightarrow (\min \{ f(a),f(c) \} \le f(b) \le \max \{ f(a),f(c) \} ) ]$

and this - it seems to me - contradicts non-monotonicity of $f$.