TITLE: Mathematical meaning of "may not"
QUESTION [5 upvotes]: Does "may not" mean that never allowed or sometimes not allowed? For example: the sequence may not converge. Does this mean that the sequence never converges or that there is no guarantee that the sequence converges?

REPLY [5 votes]: In everyday English, the construction "$X$ may not $Y$" can mean either that it is not allowed for $X$ to do $Y$, or that it is possible/conceivable that $X$ does not do $Y$. One needs to look to context and semantics to find out which of these in the case.
In mathematics it is unusual to speak about permission at all -- mathematical objects do whatever they do whether we want them to or not -- so generally the only meaning that makes sense in a mathematical context is that it is possible that $X$ does not do $Y$.