closed nest
() open nest
(], [) half-open nests
 => common point...... std theorem, sufficiency
common point => ........ not shown anywhere, necessity
Various sources state- (0,1/n) => no common point- demonstrates necessity of . That is wrong. To show necessity of  you have to show (), (], and [) never have a common point. That is not true, ie (1,1/n) => ncp and (-1/n,1/n) => cp. Therefore  is not necessary.
See post #6 for complete theorem on open and closed nests.
EDIT: Though I can access MHF from my computer again, I just noticed I wasn't notified of previous post.