This question is from Harvard seminar problem set (Q-3 part b) https://www.studocu.com/en-us/document/harvard-university/economics/mandatory-assignments/econ2020a-14-ps01-please-give-as-much-additional-information-as-possible/3513583/view
Show that it holds continuous, monotonic, LNS and transversality. $$|x_1-x_2 -y_1-y_2| \le 1$$ if x~y
I can just show transversality condition
first, for x~y $$-1\le x_1-x_2-y_1-y_2\le 1 $$ for y~z $$-1\le y_1-y_2-z_1-z_2\le 1 $$
When I sum these in equalities, I will obtain
$$-2\le x_1-x_2-z_1-z_2\le 2 $$
So, At the same time, I can say that
$$-1\le x_1-x_2-z_1-z_2\le 1$$
This implies that x~z. Thus, transversality holds. But I cannot prove for others even though I know their definitions. Please help me to do this.