query: Positive real number M such that for all n $|a_n|$ less than M
positive_passages.docid: 153121
positive_passages.text: show that $\{x_n\}$ is a cauchy sequence
negative_passages.docid: 51649
negative_passages.text: prove there is no smallest positive rational number
negative_passages.docid: 115258
negative_passages.text: Find all positive integers $n$ such that $\phi(n)+\sigma(n)=2n$.
negative_passages.docid: 156541
negative_passages.text: $a^b > b^a$ or $a^b b$ where $a,b$ are positive real numbers
negative_passages.docid: 116811
negative_passages.text: Proofs with limit superior and limit inferior: $\liminf a_n \leq \limsup a_n$
negative_passages.docid: 46039
negative_passages.text: Prove that if $x \in R,$ then there exists $n \in Z$ satisfying $x \leq n < x+1$
negative_passages.docid: 51129
negative_passages.text: Proving $n^\epsilon > \log(n)$ for sufficiently large $n$
negative_passages.docid: 157733
negative_passages.text: Are positive real numbers $x,y$ allowed to be taken out during this proof?
negative_passages.docid: 96152
negative_passages.text: Showing that $a^n - 1 \mid a^m - 1 \iff n \mid m$
negative_passages.docid: 124448
negative_passages.text: For every real number $a$ there exists a sequence $r_n$ of rational numbers such that $r_n$ approaches $a$.
negative_passages.docid: 90806
negative_passages.text: Find all positive integers $n$ such that $n$ is divisible by all the positive integers less than or equal to $\sqrt{n}$
negative_passages.docid: 78294
negative_passages.text: How does one denote the set of  all positive real numbers?
negative_passages.docid: 45939
negative_passages.text: For which $n≥2$ does there exist $\sigma\in A_n$ with $|\sigma| > n$
negative_passages.docid: 149056
negative_passages.text: Prove that for all $n\geqslant 1$ we have $F_n<{\left(\frac 74\right)}^n$.
negative_passages.docid: 148163
negative_passages.text: Show that if m/n is a good approximation of $\sqrt{2}$ then $(m+2n)/(m+n)$ is better
negative_passages.docid: 93771
negative_passages.text: Prove that for any set $A$ there is some $x \notin A$
negative_passages.docid: 11586
negative_passages.text: Sequences of Real Numbers in a closed bounded interval