query: Proving an inequality between the difference of $\sqrt2$ and any rational number.
positive_passages.docid: 144244
positive_passages.text: Bound on difference between irrational square root and rational number
negative_passages.docid: 23752
negative_passages.text: Proving Irrationality
negative_passages.docid: 91617
negative_passages.text: Prove this inequality $a \sqrt{1-b^2}+b\sqrt{1-a^2}\le1 $
negative_passages.docid: 138819
negative_passages.text: Prove that $\phi(n) \geq \sqrt{n}/2$
negative_passages.docid: 12781
negative_passages.text: Prove the following inequality $\sqrt[n]{x_1x_2...x_n}\ge (n-1)$
negative_passages.docid: 95293
negative_passages.text: Proof that $|\sqrt{x}-\sqrt{y}| \leq \sqrt{|x-y|},\quad x,y \geq 0$
negative_passages.docid: 19078
negative_passages.text: Prove an interpolation inequality
negative_passages.docid: 54369
negative_passages.text: Proof of $n^2 \leq 2^n$.
negative_passages.docid: 105047
negative_passages.text: Prove the inequality $|xy|\leq\frac{1}{2}(x^2+y^2)$
negative_passages.docid: 84558
negative_passages.text: Prove that the square root of a positive integer is either an integer or irrational
negative_passages.docid: 77026
negative_passages.text: Proof with inequalities
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: 158276
negative_passages.text: Inequality problem involving square roots
negative_passages.docid: 15237
negative_passages.text: Prove an inequality using complex analysis
negative_passages.docid: 146180
negative_passages.text: Prove that $x\sqrt{1-x^2} \leq \sin x \leq x$
negative_passages.docid: 88838
negative_passages.text: How can I prove this inequality for $n\geq 2$?
negative_passages.docid: 28218
negative_passages.text: Elementary proof for $\sqrt[3]{xyz} \leq \dfrac{x+y+z}{3}$