 And in fact, if we take the definition of a Dedekind cut, and relax the requirement that the left and right sets be non-empty, we get two new elements—the one with an empty right set is greater than every rational number, and called ‘\(+\infty\)’, while the one with an empty left set is less than every rational number, and called ‘\(-\infty\)’. Within these ‘extended real numbers’, not only does every bounded increasing sequence have a limit, but every increasing sequence has a limit.