I'm having troubles to prove that $f(x+y)=f(x)+f(y)$
Let $K$ be a complete ordered field and $0'$ and $1'$ be the zero and the unit of $K$. For each $n\in \mathbb N$, we have $n'=n\cdot1'=+1'+...+1$ (n-times) and $(-n)'=-n'$. We define a function $f:\mathbb R\ \to K$ with $f(p/q)=p'/q'$ for each $p/q\in \mathbb Q$ and for each x irracional, $f(x)=\sup\{p'/q'\in K;p/q\lt x\}$. Prove that $f$ is a homomorphism.
Since
$\{p'/q'\in K;p/q\lt x\}+ \{p'/q'\in K;p/q\lt y\}\subset \{p'/q'\in K;p/q\lt x+y\}$, we have $f(x+y)\geq f(x)+f(y)$. I need help to prove the inverse, i.e., $f(x+y)\leq f(x)+f(y)$.
I really need help.
Thanks a lot.