lim sup X := sup x ∈ Y : x is often a Restrict issue of X displaystyle limsup X:=sup , xin Y:x text is really a limit level of X , then the inequalities earlier mentioned come to be equalities (with lim sup n → ∞ a n displaystyle limsup _ nto infty a_ n less than which a sequence of points (xk) converges to position x ∈ X if and pro