Nest theorems never explain why nests have to be closed. Does this float?

Given nests (ai,bi), (ai,bi], [ai,bi] and limai=limbi=a. Assume x belongs to all intervals. Then:

lim(ai<x<bi)=(a<x<a) => no x in (ai,bi)

lim(ai<x$\displaystyle \leq$bi)=(a<x$\displaystyle \leq$a)=> no x in (ai,bi]

lim(ai$\displaystyle \leq$x$\displaystyle \leq$bi)=(a$\displaystyle \leq$x$\displaystyle \leq$a) => x=a in [ai,bi]

Ex

(-1/n,1/n) no x

(-1/n,1/n] no x

[-1/n,1/n] x=0