Answer: First, a singleton set is the set with only one element, for example, the set {∅} or set "null" is a singleton.
now, X ≥ 0, and X ∈ R.
(x,x + 1]∩N is the intersection between the set x and x+1 with the set of natural numbers, so the result of this operation is the set of all the natural numbers between x and x+1.
if x is a rational (or irrational) number, such as 1.5, then x +1 = 2.5
in (1.5, 2.5] there is only one natural element, the 2, then (x,x + 1]∩N is the singleton {2}
if x is a positive integer number, such as 1, then x + 1 = 2
the set is (1,2] is important to notice that 1 ∉ (1,2] because the set is open in the lower limit. so this set only contains one natural number, the 2; so again; the intersection with the natural set is a singleton
So this is proven for x rational, irrational and integers.
This happens because each element in the natural set is separated by the next one by 1, and in the set (x,x + 1] although the distance between the lower and upper limits is 1, the set is open in the lower limit, so the distance is < 1. then you never can have two natural numbers on this set.
