# Bounding a sum of real functions with rationals

• Oct 14th 2011, 03:46 AM
akolman
Bounding a sum of real functions with rationals
I am stuck with this problem

Show that for real functions $f$ and $g$ such that $f+g, there exists rationals $r$ and $s$ such that $r+s and $g

Would using the Archimedean principle help?

Let $\alpha \in \mathbb{Q}$ and $\alpha>0$.
By the Archimedean principle, there exists $k \in \mathbb{Z}$ such that
$k\alpha \leq x < (k+1)\alpha$

Is this in the right direction?

• Oct 14th 2011, 04:40 AM
Georgii
Re: Bounding a sum of real functions with rationals
Note that $f. So, there exists rational $r$, such that $f... And now think how to obtain second rational
• Oct 14th 2011, 06:57 AM
Hartlw
Re: Bounding a sum of real functions with rationals
"Show that for real functions $f$ and $g$ such that $f+g, there exists rationals $r$ and $s$ such that $r+s and $g"

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