query: Does there exist $\tau \in \mathbb{R}$ such that $a_n - n\tau$ is bounded?
positive_passages.docid: 108425
positive_passages.text: Problem on abelian group
negative_passages.docid: 80920
negative_passages.text: Prove or disprove: For any $v\in\mathbb{R}^n, \|v\|_1\|v\|_\infty \leq \frac{1+\sqrt{n}}{2}\|v\|_2^2$
negative_passages.docid: 120648
negative_passages.text: Proof Check: Every Cauchy Sequence is Bounded
negative_passages.docid: 31255
negative_passages.text: Bounded sequence with divergent Cesaro means
negative_passages.docid: 23231
negative_passages.text: Convergence of $A_nT$ to $AT$ in operator norm for compact $T$
negative_passages.docid: 124959
negative_passages.text: Does weak compactness imply boundedness in a normed vector space (not necessarily complete)?
negative_passages.docid: 130877
negative_passages.text: Boundedness and Cauchy Sequence: Is a bounded sequence such that $\lim(a_{n+1}-a_n)=0$ necessarily Cauchy?
negative_passages.docid: 147021
negative_passages.text: Proving a function is bounded
negative_passages.docid: 57868
negative_passages.text: Prove $\tau$ is cofinite on $X$ iff $(X,\tau)$ is $T_1$-space and any infinite subset of $X$ is dense in $(X, \tau)$
negative_passages.docid: 19741
negative_passages.text: If $(\mu_t)_{t\in I}$ and $(\mu_t\ast\nu_t)_{t\in I}$ are tight, then $(\nu_t)_{t\in I}$ is tight
negative_passages.docid: 28468
negative_passages.text: Haar measure on $SO(n)$
negative_passages.docid: 54321
negative_passages.text: Questions about $f: \mathbb{R} \rightarrow \mathbb{R}$ with bounded derivative
negative_passages.docid: 86319
negative_passages.text: Give an example of a bounded, non-convergent real sequence $(a_n)$ s.t. $a_n-a_{n-1}\rightarrow 0$
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: 3896
negative_passages.text: A closed set in $(A, \mathcal T_A)$ that is not closed in $(X, \mathcal T)$
negative_passages.docid: 60520
negative_passages.text: Upper bound for $|f(0)|$ based on its behavior within the unit disk.
negative_passages.docid: 35547
negative_passages.text: Does $\{f_ng_n\}\to fg$ uniformly?