query: Every irrational number is the limit of a sequence of rational numbers
positive_passages.docid: 143800
positive_passages.text: continuous map on $\mathbb{R}$ which is the identity on $\mathbb{Q}$ is the identity map, hence Aut$(\mathbb{R}/\mathbb{Q})= 1.$
negative_passages.docid: 30478
negative_passages.text: How to prove that a number is irrational
negative_passages.docid: 5797
negative_passages.text: Limit of the sequence $a_n=\sqrt[n+1]{(n+1)!}-\sqrt[n]{n!}$
negative_passages.docid: 77129
negative_passages.text: Density of irrationals
negative_passages.docid: 142000
negative_passages.text: Prove that there is an irrational number and a rational number between any two distinct real numbers
negative_passages.docid: 87116
negative_passages.text: Prove $ \lim_{n\to \infty}\frac{n}{n+1} = 1$ using Cauchy (epsilon) definition of a limit of a sequence
negative_passages.docid: 63894
negative_passages.text: These two sequences have the same limit
negative_passages.docid: 89814
negative_passages.text: Limit of a Piecewise Function defined on Rationals and Irrationals
negative_passages.docid: 120648
negative_passages.text: Proof Check: Every Cauchy Sequence is Bounded
negative_passages.docid: 98666
negative_passages.text: Continuity defined for rational and irrational numbers
negative_passages.docid: 29266
negative_passages.text: Limit of a sequence defined by recursive relation : $ a_n = \sqrt{a_{n-1}a_{n-2}}$
negative_passages.docid: 148035
negative_passages.text: Is it possible for a sequence of rational numbers to have a non rational limit?
negative_passages.docid: 67420
negative_passages.text: Limit of quotient sequence
negative_passages.docid: 60024
negative_passages.text: Proof that a sequence is convergent
negative_passages.docid: 92734
negative_passages.text: Existence of a sequence of continuous functions convergent pointwise to the indicator function of irrational numbers
negative_passages.docid: 31867
negative_passages.text: Elementary proof that the limit of $\sum_{i=1}^{\infty} \frac{1}{\operatorname{lcm}(1,2,...,i)}$ is irrational
negative_passages.docid: 154578
negative_passages.text: Proving there is a sequence convergent to a limit point of a set without axiom of countable choice?