query: Let x be real. Show that, for any ϵ>0, there exist two rationals such that q<x<q′ and |q−q′|<ϵ
positive_passages.docid: 148585
positive_passages.text: Show that, for any $\epsilon>0$, there exist two rationals such that $q < x < q'$ and $|q-q'|<\epsilon$
negative_passages.docid: 61472
negative_passages.text: If $0<x<y$, then prove that $\sqrt{x} <\sqrt{y}$ and $x <\sqrt{xy} <y$
negative_passages.docid: 46831
negative_passages.text: Show that $ a，b, \sqrt{a}+ \sqrt{b} \in\mathbb Q \implies \sqrt{a},\sqrt{b} \in\mathbb Q $
negative_passages.docid: 72313
negative_passages.text: Show that for all real numbers $a$ and $b$, $\,\, ab \le (1/2)(a^2+b^2)$
negative_passages.docid: 38851
negative_passages.text: To find the probability whether $X$ is rational
negative_passages.docid: 23752
negative_passages.text: Proving Irrationality
negative_passages.docid: 125707
negative_passages.text: A simple proof (hopefully) for $2^x \neq 0$
negative_passages.docid: 63409
negative_passages.text: If $x^2+ax+b=0$ has a rational root, show that it is in fact an integer
negative_passages.docid: 64905
negative_passages.text: Let $f \in C [a,b]$ such that $\forall x\exists y$ such that $|f(y)| \leq \frac{1}{2}|f(x)|$. Prove that $\exists c\in [a,b]$ such that $f(c) = 0$.
negative_passages.docid: 5410
negative_passages.text: Prove that $a^2 + b^2 \geq 8$ if $ x^4 + ax^3 + 2x^2 + bx + 1 = 0 $ has at least one real root.
negative_passages.docid: 56877
negative_passages.text: Show that $\sqrt{4 + 2\sqrt{3}} - \sqrt{3}$ is rational.
negative_passages.docid: 93410
negative_passages.text: How to prove ~ (~P & ~Q) : P ∨ Q by natural deduction
negative_passages.docid: 140817
negative_passages.text: Real Analysis Inequality Proof Involving Reals and Rationals $0 < |r - q| < \varepsilon$
negative_passages.docid: 114872
negative_passages.text: If $ax^2-bx+c=0$ has two distinct real roots lying in the interval $(0,1)$ $a,b,c$ belongs to natural prove that $\log_5 {abc}\geq2$
negative_passages.docid: 156766
negative_passages.text: Prove that $\sqrt{\frac{n}{n+1}}\notin \mathbb{Q}$
negative_passages.docid: 34998
negative_passages.text: How to prove that $\mathbb{Q}$ ( the rationals) is a countable set
negative_passages.docid: 106401
negative_passages.text: Prove: $p-mq | f(m)$ where $m$ is any integer