Bounding a sum of real functions with rationals

I am stuck with this problem

Show that for real functions and such that , there exists rationals and such that and

Would using the Archimedean principle help?

Let and .

By the Archimedean principle, there exists such that

Is this in the right direction?

Thanks in advance

Re: Bounding a sum of real functions with rationals

Note that . So, there exists rational , such that ... And now think how to obtain second rational

Re: Bounding a sum of real functions with rationals

"Show that for real functions and such that , there exists rationals and such that and "

let A =lubf, B=lubg

If A and B are rational and belong to f and g respectiveley, you are finished.

Otherwise you can pick rational numbers r and s greater than A and B respectiveley and arbitrarily close so that r+s < x