query: Show that the sum of the oscilations is less or equal to $f(b)-f(a)$
positive_passages.docid: 85465
positive_passages.text: Let $f$ be increasing on $[a,b]$ and $a < x_1 < \dotsb < x_n < b$. Show that $\sum_{k = 1}^n [f(x_k^+) - f(x_k^-)]\leq f(b^-) - f(a^+)$.
negative_passages.docid: 48092
negative_passages.text: Inequality involving a finite sum
negative_passages.docid: 85626
negative_passages.text: prove $\sum_{i=1}^{n}(x_i-\bar{x})^2\lt\sum_{i=1}^{n}(x_i-a)^2$
negative_passages.docid: 131692
negative_passages.text: How to show that $f_{0}f_{1}+f_{1}f_{2}+...+f_{2n-1}f_{2n} = f^{2}_{2n}$
negative_passages.docid: 31522
negative_passages.text: Proving $f$ has at least one zero inside unit disk
negative_passages.docid: 134812
negative_passages.text: Proving that if $\sum\|f_n-e_n\|^2< 1$, $\{f_n\}$ is a complete sequence
negative_passages.docid: 18046
negative_passages.text: Prove the following equality: $\sum_{k=0}^n\binom {n-k }{k} = F_n$
negative_passages.docid: 145884
negative_passages.text: Showing $\left|\frac{a+b}{2}\right|^p+\left|\frac{a-b}{2}\right|^p\leq\frac{1}{2}|a|^p+\frac{1}{2}|b|^p$
negative_passages.docid: 78319
negative_passages.text: Show that $x \mapsto \frac{f(x)}{x}$ is strictly increasing on (0,1) given that f '(x) is strictly increasing on (0,1) and that f(0)=0
negative_passages.docid: 53024
negative_passages.text: $f(x)f(1/x)=f(x)+f(1/x)$
negative_passages.docid: 48642
negative_passages.text: Let $f(n) = \sum_{(k,n)=1} e^\frac{2\pi ik}{n}$. Show that $f(n) = \sum_{k=1}^{n}\left(\sum_{d\mid (k,n)} \mu(d)e^{\frac{2\pi ik}{n}} \right) $
negative_passages.docid: 156513
negative_passages.text: Prove that inequality $\sum _{cyc}\frac{a^2+b^2}{a+b}\le \frac{3\left(a^2+b^2+c^2\right)}{a+b+c}$
negative_passages.docid: 119552
negative_passages.text: Summing Finitely Many Terms of Harmonic Series: $\sum_{k=a}^{b} \frac{1}{k}$
negative_passages.docid: 149739
negative_passages.text: Prove: $f(t)\leq t+1$
negative_passages.docid: 39921
negative_passages.text: show that f is constant on I
negative_passages.docid: 157328
negative_passages.text: A complex analysis problem to prove an inequality
negative_passages.docid: 131000
negative_passages.text: A proof of a criterion of vanishing of a function.