query: Does $\forall a\neq 0,N(a)>N(0)$ imply $N$ is a Euclidean function?
positive_passages.docid: 17161
positive_passages.text: Can anyone make me understand in simple language why the second condition for being an Euclidean domain is superfluous?
negative_passages.docid: 40797
negative_passages.text: $(a_n+a_{n+1})$ is convergent implies $(a_n/n)$ converges to $0$
negative_passages.docid: 139652
negative_passages.text: summable square function implies...?
negative_passages.docid: 67214
negative_passages.text: Function $ y = x^n - nx$, where $x \geq 0$ and $n \geq 2$.
negative_passages.docid: 84969
negative_passages.text: When does $\lim_{n\rightarrow \infty} a_n = 0$ imply $\sum_{n = 0}^{\infty} a_n$ converges?
negative_passages.docid: 158730
negative_passages.text: Is $\ln(\ln(n))$ irrational for any integer $n>1$?
negative_passages.docid: 126821
negative_passages.text: Let $f:[0,1]\to \mathbb{R}$. Show that $\{a \in [0,1]: f(a)\neq 0\} = \bigcup_{n=1}^{\infty} \{a \in [0,1]: |f(a)|\geq 1/n\}$
negative_passages.docid: 12031
negative_passages.text: Asymptotic behavior of the sequence $a_n=\sqrt{n+a_{n-1}}$
negative_passages.docid: 53443
negative_passages.text: Does $a_n \sim b_n$ imply $\sum_n a_n \sim \sum_n b_n$ for $a_n, b_n>0$?
negative_passages.docid: 138178
negative_passages.text: Proving the number of iterations in the Euclidean algorithm
negative_passages.docid: 88790
negative_passages.text: Positive harmonic function on $\mathbb{R}^n$ is a constant?
negative_passages.docid: 89325
negative_passages.text: $a_{m+n}\leq a_ma_n$ implies the convergence of $\sqrt[n]a_n$
negative_passages.docid: 103550
negative_passages.text: Multiplicative Euclidean Function for an Euclidean Domain
negative_passages.docid: 135018
negative_passages.text: Proving that a sequence has $a_n = a_{n+2}$ for $n$ sufficiently large.
negative_passages.docid: 14409
negative_passages.text: A problem about function N
negative_passages.docid: 61858
negative_passages.text: Is there $\phi(n)=n/6$
negative_passages.docid: 53001
negative_passages.text: If $|f'(c)| \leq M$ and $ M < 1$, the sequence defined by $a_{n+1} = f(a_n)$ converges