"For any [real number] x and positive integer N, there is an integer n such that ." (see ita-2-3-14-pg26)
"To show this we merely have to apply LUB3 to the number Nx, getting an integer n such that ." (see ita-2-3-14-pg26)
Note that we can't use O3 to get from Rosenlicht's intermediate result, , to the final desired result (by multiplying both sides of the inequalities by 1/N) because we are missing the greater than zero condition. Instead, please see my first reading of LUB4 here.