query: Let $x_{n}$ be a monotonously decreasing sequence converging to $0$. prove $nx_{n}$ is also convergent to $0$.
positive_passages.docid: 16389
positive_passages.text: If $(a_n)\subset[0,\infty)$ is non-increasing and $\sum a_n<\infty$, then $\lim{n a_n} = 0$
negative_passages.docid: 83040
negative_passages.text: Proving that the sequence $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$ is boundedly convergent on $\mathbb{R}$
negative_passages.docid: 133451
negative_passages.text: To show that ${x_n}$ is convergent where $|x_{n+1} - x_n|< \frac{1}{n^2}$
negative_passages.docid: 107311
negative_passages.text: Convergence of a sequence and some of its subsequences
negative_passages.docid: 129393
negative_passages.text: Product of bounded and convergent to $0$ sequence is a convergent to $0$ sequence
negative_passages.docid: 66951
negative_passages.text: Proving the existence of a sequence of functions converging to 0 with an added property
negative_passages.docid: 55464
negative_passages.text: Convergence of sequence given by $x_1=1$ and $x_{n+1}=x_n+\sqrt{x_n^2+1}$
negative_passages.docid: 152525
negative_passages.text: If $b_n$ is a bounded sequence and $\lim a_n = 0$, show that $\lim(a_nb_n) = 0$
negative_passages.docid: 132948
negative_passages.text: Prove if $\left\{x_n\right\}$ converges to $2$, then $\left\{\frac{1}{x_n}\right\}$ converges to $\frac{1}{2}$
negative_passages.docid: 104344
negative_passages.text: Prove if $|a_{n+1}|/|a_n| \leq |b_{n+1}|/|{b_n}|$ and $\sum |b_n|$ is convergent, then $\sum |a_n|$ is convergent
negative_passages.docid: 46778
negative_passages.text: Prove that a recursive sequence is monotonically increasing
negative_passages.docid: 46323
negative_passages.text: Criteria for convergent sequence (Baby Rudin Theorem 3.22)
negative_passages.docid: 27204
negative_passages.text: Proof for convergence of a given progression $a_n := n^n / n!$
negative_passages.docid: 154553
negative_passages.text: If $a_k\geq0$ is a decreasing sequence converging to $0$, do we have $\lim_{n\to\infty}\frac{\sum_{k=1}^n a_k}n = 0$?
negative_passages.docid: 100580
negative_passages.text: Proof that a sequence converges.
negative_passages.docid: 63562
negative_passages.text: Show that $x_{n+1}-\dfrac{x_n}{2}$ converges to zero implies that $(x_n)_n$ also converges to zero.
negative_passages.docid: 134261
negative_passages.text: for every convergent sequence $x_n$, $f(x_n)$ also converges. Does this imply continuity of f?