Background:
Definition: An integral domain $R$ is a Euclidean domain if there is a function $\delta$ from the nonzero elements of $R$ to the nonnegative integers with these properties;
(i) If $a$ and $b$ are nonzero elements of $R,$ then $\delta(a)\leq \delta(ab).$
(ii) If $a,b\in R$ and $b\neq 0_R,$ then there exist $q,r\in R$ such that $a=bq+r$ and either $r=0_R$ or $\delta(r) < \delta(b).$
Exercise 13 a: Let $R$ be a Eucidean domain with function $\delta$ and let $k$ be a positive integer.
(a) Show that $R$ is also a Euclidean domain under the function $\theta$ given by $\theta(r)=\delta(r)+k.$
Proof: It is easy to check that $\theta(ab)=\delta(ab)+k\leq \delta(a)\delta(b)+k\leq(\delta(a)+k)(\delta(b)+k)=\theta(ab)$. Now suppose $a=bq+r$ where $0\leq r<|b|$. where either $r=0$ or $\delta(r)<\delta(b)$. In the case $r\neq 0$ check $\theta(a)=\delta(a)+k\leq \delta(b)+k=\theta(b)$.
Questions:
For exercise 13a) above, to show (i) of the definition of Euclidean domain. I at first did the following;
Let $a,b\in R$ and let $k$ be a fixed integer. with $a<b$, $a<ab$, $\delta(a)+k<\delta(b)+k$, and $\delta(a)+k<\delta(ab)+k$. Then $\theta(a)=\delta(a)+k<\delta(ab)+k=\theta(ab)$. I also know that $\delta(a)\delta(b)\leq \delta(a)(\delta(b)+k)\leq (\delta(a)+k)(\delta(b)+k)=\theta(a)\theta(b)$.
In the solution above for the exercise above for showing (i) of defintion for Euclidean domain, part of it says: $\theta(ab)=\delta(ab)+k\leq \delta(a)\delta(b)+k\leq (\delta(a)+k)(\delta(b)+k)=\theta(a)\theta(b)$. I don't understand how $\delta(ab)+k\leq \delta(a)\delta(b)+k$ is valid when it has not know shown $\delta(ab)\leq \delta(a)\delta(b)$. I am not sure if there is a typo in the step: $\delta(a)\delta(b)+k\leq (\delta(a)+k)(\delta(b)+k)$. It seems it should be $\delta(a)(\delta(b)+k)$ instead of $\delta(a)\delta(b)+k$. Also, why is showing $\theta(ab)=\theta(a)\theta(b)$ needed?
Thank you in advance