query: Proof that $|\sin(nx)| \le n|\sin(x)|$
positive_passages.docid: 150819
positive_passages.text: How to prove by induction that $|\sin(nx)| \leq n|\sin x|$?
negative_passages.docid: 85944
negative_passages.text: Prove $ \sin x + \frac{ \sin3x }{3} + ... + \frac{ \sin((2n-1)x) }{2n-1} >0 $
negative_passages.docid: 149889
negative_passages.text: Prove that $|\log(1 + x^2) - \log(1 + y^2)| \le |x-y|$
negative_passages.docid: 118613
negative_passages.text: Proving an inequality: $|1-e^{i\theta}|\le|\theta|$
negative_passages.docid: 129890
negative_passages.text: Proving inequality $|\sin(n\theta) |\le n\sin\theta$
negative_passages.docid: 48586
negative_passages.text: Prove: $\cot x=\sin x\,\sin\left(\frac{\pi}{2}-x\right)+\cos^2x\,\cot x$
negative_passages.docid: 49054
negative_passages.text: Show that $\left | \sin a - \sin b \right | \leq \left | a-b \right |$
negative_passages.docid: 70715
negative_passages.text: How do I prove $\int_0^\pi\frac{(\sin nx)^2}{(\sin x)^2}dx = n\pi$?
negative_passages.docid: 40420
negative_passages.text: Proof by induction of Bernoulli's inequality: $(1 + x)^n \geq 1 + nx$
negative_passages.docid: 39073
negative_passages.text: Proving that: $\forall n \in \mathbb{N} :|f^{(n)}(x)|\leq \frac{1}{n+1}$
negative_passages.docid: 113593
negative_passages.text: How can I prove that $\frac{n^2}{x_1+x_2+\dots+x_n} \le \frac{1}{x_1}+ \frac{1}{x_2} +\dots+ \frac{1}{x_n}$?
negative_passages.docid: 146180
negative_passages.text: Prove that $x\sqrt{1-x^2} \leq \sin x \leq x$
negative_passages.docid: 90896
negative_passages.text: proof of inequality $e^x\le x+e^{x^2}$
negative_passages.docid: 70304
negative_passages.text: Prove there are $\xi$, $\eta$ with $f'(\xi)f'(\eta)=1$
negative_passages.docid: 52614
negative_passages.text: (exercise from Tao's analysis book) Proof of a lemma relating to power set of X
negative_passages.docid: 38185
negative_passages.text: Prove that $\{\cos(nx) : n=1, ... ,\infty \}$ is bounded but not totally bounded in $C([0,\pi])$
negative_passages.docid: 1588
negative_passages.text: How to prove $\int_{-\pi}^\pi (f(x))^2dx\le \int_{-\pi}^\pi (f'(x))^2dx$