Is every strictly locally monotone sequence strictly monotone?

Question

sequence b(sub n), n=1,2,3... is strictly locally monotone if for every integer k>1 either b(sub k-1) < b(sub k) < b(sub k-1) OR b(sub k-1) > b(sub k) >b(sub k+1).
Prove it is strictly monotone or give counterexample.

Answer 1

It appears that a sequence would satisfy your defn of locally monotone if, for example, b_1 < b_2 < b_3 and b_4 > b_5 > b_6, but such a sequence is not monotone.

Answer 2

i think it is strictly monotone, use induction