query: if ${m/n}$ is an approximation to $\sqrt{2}$, prove $m/2n+n/m$ is always a better approximation.
positive_passages.docid: 142768
positive_passages.text: Approximation of $\sqrt{2}$
negative_passages.docid: 67605
negative_passages.text: Different proof that $\sqrt{2}$ is irrational
negative_passages.docid: 100375
negative_passages.text: Prove that ${\sqrt {n} }^{\sqrt {n+1}} > {\sqrt {n+1}}^{\sqrt {n}}$.
negative_passages.docid: 129433
negative_passages.text: $\sqrt{m_1}+\sqrt{m_2}+ \cdots + \sqrt{m_n}$ is Irrational
negative_passages.docid: 14581
negative_passages.text: How do we prove $n^n \mid m^m \Rightarrow n \mid m$?
negative_passages.docid: 121694
negative_passages.text: How to prove $\log n \leq \sqrt n$ over natural numbers?
negative_passages.docid: 95000
negative_passages.text: Using induction to prove that $2 \mid (n^2 − n)$ for $n\geq 1$
negative_passages.docid: 140841
negative_passages.text: Induction proof of lower bound for $\sum \sqrt n$
negative_passages.docid: 33786
negative_passages.text: Proof that a certain fraction is always an integer
negative_passages.docid: 6582
negative_passages.text: Prove the existence of the square root of $2$.
negative_passages.docid: 54369
negative_passages.text: Proof of $n^2 \leq 2^n$.
negative_passages.docid: 91101
negative_passages.text: Proving $\sqrt2$ is irrational
negative_passages.docid: 72470
negative_passages.text: Prove that $2|ab| \leq a^2 + b^2$ and $|a|+|b| \leq \sqrt {2}(a^2+b^2)^{1/2}$
negative_passages.docid: 107142
negative_passages.text: If $(x_n) \to x$ then $(\sqrt[n]{x_1x_2\cdots x_n}) \to x$
negative_passages.docid: 95626
negative_passages.text: $\int_X |f_n - f| \,dm \leq \frac{1}{n^2}$ for all $n \geq 1$ $\implies$ $f_n \rightarrow f$ a.e.
negative_passages.docid: 125267
negative_passages.text: Prove that $2^{1/2}$ is irrational
negative_passages.docid: 17289
negative_passages.text: prove $\sqrt{a_n b_n}$ and $\frac{1}{2}(a_n+b_n)$ have same limit