query: How can I prove that if $a < b$ then $a^2$ < $b^2$ for $a,b \in \Bbb N$
positive_passages.docid: 113168
positive_passages.text: If $a > b$, is $a^2 > b^2$?
negative_passages.docid: 20626
negative_passages.text: Prove that $x^y < y^x$
negative_passages.docid: 137511
negative_passages.text: Trying to show that $\sup \{b^p:p\in Q,\;0<p<x\} = \inf\{b^q:q\in Q,\;x<q\}$
negative_passages.docid: 85731
negative_passages.text: Prove by induction that $n! > n^2$
negative_passages.docid: 153313
negative_passages.text: Prove that for all integers $ a $ and $ b $, if $ a $ divides $ b $, then $ a^{2} $ divides $ b^{2} $.
negative_passages.docid: 84946
negative_passages.text: If $0<a<b$, prove that $a<\sqrt{ab}<\frac{a+b}{2}<b$
negative_passages.docid: 22073
negative_passages.text: Prove by induction that $n^2<n!$
negative_passages.docid: 64782
negative_passages.text: How to prove if $n$ is prime and $n | a^2$ then $n | a$?
negative_passages.docid: 26741
negative_passages.text: How to prove for $s<1,|a+b|^s\le|a|^s+|b|^s$
negative_passages.docid: 143130
negative_passages.text: Proving that the supremum of $(a,b)$ is $b$.
negative_passages.docid: 36602
negative_passages.text: $a^m=b^m$ and $a^n=b^n$ imply $a=b$
negative_passages.docid: 92975
negative_passages.text: Does $a^n \mid b^n$ imply $a\mid b$?
negative_passages.docid: 110256
negative_passages.text: Let $a,b \in {\mathbb{Z_+}}$ such that $a|b^2, b^3|a^4, a^5|b^6, b^7|a^8 \cdots$, Prove $a=b$
negative_passages.docid: 48081
negative_passages.text: Prove that for all $n \in \mathbb{N}$: $2^{n-1}(a^n + b^n) \ge (a + b)^n$
negative_passages.docid: 101872
negative_passages.text: Prove that $\gcd(a^2, b^2) = \gcd(a, b)^2$
negative_passages.docid: 130281
negative_passages.text: Proving $a^ab^b + a^bb^a \le 1$, given $a + b = 1$
negative_passages.docid: 54369
negative_passages.text: Proof of $n^2 \leq 2^n$.