query: Show that for any $x,y \in X$, $\sum_{k=1}^\infty|\left\left|\le \|x\|\|y\|.$
positive_passages.docid: 27901
positive_passages.text: Proving $\sum\limits_{i=1}^k | \langle x,v_i \rangle \langle y,v_i\rangle| \leq \|x\|\cdot \|y\|$
negative_passages.docid: 1525
negative_passages.text: If for any $k$, $\sum\limits_{n=0}^\infty a_n^k=\sum\limits_{n=0}^\infty b_n^k$, then$(a_n)=(b_{σ(n)}),\ σ \in{\mathfrak S}_{\mathbb N}$
negative_passages.docid: 8913
negative_passages.text: Show that there is $\xi$ s.t. $f(\xi)=f\left(\xi+\frac{1}{n}\right)$
negative_passages.docid: 5765
negative_passages.text: Show that $[2x]+[2y] \geq [x]+[y]+[x+y]$
negative_passages.docid: 98941
negative_passages.text: Closed formula for the sums $\sum\limits_{1 \le i_1 < i_2 < \dots < i_k \le n} i_1 i_2 \cdots i_k $?
negative_passages.docid: 154703
negative_passages.text: Use combinatorics to prove $L\left(n,k\right)=\sum_{j=0}^{n}{ n\brack j}{j\brace k}$
negative_passages.docid: 29679
negative_passages.text: Prove: $|x-y|\leq |x|+|y|$
negative_passages.docid: 104447
negative_passages.text: Prove that $\sqrt{n} \le \sum_{k=1}^n \frac{1}{\sqrt{k}} \le 2 \sqrt{n} - 1$ is true for $n \in \mathbb{N}^{\ge 1}$
negative_passages.docid: 104365
negative_passages.text: Evaluate: $\lim_{n\to\infty}\left({2\sqrt n}-\sum_{k=1}^n\frac1{\sqrt k}\right)$
negative_passages.docid: 102942
negative_passages.text: Compute the sum $\sum_{k=1}^{\infty}k^mz^k$ where $|z|<1$
negative_passages.docid: 127047
negative_passages.text: How to show that $\sum_{i=1}^n | \langle f, f_i\rangle |^2 \leq \Vert f \Vert^2$
negative_passages.docid: 11435
negative_passages.text: Show that $\sum_{k=1}^\infty \frac{\ln{k}}{k^2}$ converges or diverges
negative_passages.docid: 130553
negative_passages.text: How do I prove the following inequality $\sum_{k=n+1}^{\infty}\frac{1}{k^{2}\log k}\leq\frac{1}{n\log n}$?
negative_passages.docid: 47323
negative_passages.text: Show that $\left(1+\frac xk\right)^k$ is monotonically increasing
negative_passages.docid: 67454
negative_passages.text: Show that $\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y}) = \sum_{i=1}^n (x_i - \bar{x})(y_i)$.
negative_passages.docid: 108010
negative_passages.text: Evaluating $\sum_{k=1}^{\infty}\left(\frac{\sin(tk)}{k}\right)^2$
negative_passages.docid: 124229
negative_passages.text: Simplify $\sum_{k=0}^{\infty} x^ky^{\frac{k(k+1)}{2}}$