Given a Euclidean domain with Euclidean function $\delta$ is it possible to bound $\delta(r+r')$ in terms of $\delta(r)$ and $\delta(r')$?

Question

I came across this while thinking about another Question. Essentially everything is said in the title:

Let $R$ be a Euclidean domain and let $\delta$ be a Euclidean function for this domain. Is $\sup\lbrace\delta(r+r')\mid\delta(r),\delta(r')<n\rbrace <\infty$ for all $n\in\mathbb{N}$? Similarly one can ask weather $\sup\lbrace\delta(rr')\mid\delta(r),\delta(r')<n\rbrace <\infty$ for all $n$.

For the common euclidean domains $(\mathbb{K}[t], \mathbb{Z}[i], ...)$ this is easy to show, but at the moment i have no real idea how to generalize this.