I've found the infinite-product tag but no infinite-sum tag.

In the context of an abstract formalisation of division with remainder in an interactive theorem prover, I came upon the following problem: The following two properties seem rather natural for a division with remainder (for $b\neq 0$ and $c\neq 0$): $$(a + c \cdot b)\ \mathrm{div}\ b = c + a\ \m...

Let $R$ be an integral domain containing a prime element $p$ such that $\cap_{n \ge 1} (p^n)=(0)$ ; if $f : R \setminus \{0\} \to \mathbb Z$ defined as $f(x):=\max \{i : x \in (p^i)\}$ is a function such that $f(x)\ge 0 , \forall 0\ne x\in R$ and for every $a,b \in R$ with $ 0\ne b , a \notin...

Define a function $\delta : \mathbb{Z}[i] \rightarrow \mathbb{N}$ by $\delta(a+bi)=a^2 +b^2$. Show that $\delta$ is a Euclidean valuation on the Gaussian Integers $\mathbb{Z}[i]$ i.e. For Gaussian Integers $a,b$ there exists Gaussian Integers $q,r$ such that $a=qb+r$ with either $r=0$ or $\delta(r)<

I'm having some trouble proving that the Gaussian Integer's ring ($\mathbb{Z}[ i ]$) is an Euclidean domain. Here is what i've got so far. To be a Euclidean domain means that there is a defined application (often called norm) that verifies this two conditions: $\forall a, b \in \mathbb{Z}[i] \...

Define subsets $\Delta$ and $D$ of $\Bbb R^{2}$ by \begin{align*} \Delta &= \{ (s,t) \mid s \geq 0,\ t \geq 0,\ s+t \leq 1 \}, \\ D &= \{ (x,y) \mid x \geq 0,\ y \geq 0,\ f(x,y) \geq 0 \}. \end{align*} Let $f(x,y)=x^{2}+y^{2}+1-2xy-2x-2y$. 1. Prove that the restriction $\phi |_{\Delta}$ of ...

I am working through the proof of the following theorem in Sohr - The Navier Stokes Equations: Let $\Omega \subseteq \mathbb{R}^n$ be an arbitrary domain with $n \geq 2$. Then there exists a sequence $(\Omega_j)_{j = 1}^\infty$ of bounded Lipschitz subdomains of $\Omega$ and a sequence $(\...