It is closed! Addition is abinaryoperation. When we write we only use two numbers at once. When we write we perform addition repeately. But it is still binary i.e. we add them one at a time. But when we have then it is alimitof a finite sum i.e. the limit of . And limits have nothing to do with group theory. Thus, just because rationals are not closed under infinite sums (limit of sums) does not mean they are not closed under addition.