query: If $a_n$ is a sequence in $X$ and $a\in X$ such that $\frac{d(a_n,a)}{1+d(a_n,a)} \to 0$, then $a_n\to a$.
positive_passages.docid: 9355
positive_passages.text: If $d(x,y)$ is a metric, then $\frac{d(x,y)}{1 + d(x,y)}$ is also a metric
negative_passages.docid: 73176
negative_passages.text: $\{a_n\}$ be a sequence such that $ a_{n+1}^2-2a_na_{n+1}-a_n=0$, then $\sum_1^{\infty}\frac{a_n}{3^n}$ lies in...
negative_passages.docid: 20909
negative_passages.text: For each positive integer $n$, let $x_n=1/(n+1)+1/(n+2)+\cdots+1/(2n)$. Prove that the sequence $(x_n)$ converges.
negative_passages.docid: 64458
negative_passages.text: Prove that if $a_n\gt 0$ and $\sum a_n$ diverges, then $\sum \frac{a_n}{1+a_n}$ diverges.
negative_passages.docid: 46685
negative_passages.text: Show that the sequence $a_1=1$, $a_2=2$, $a_{n+2} = (a_{n+1}+a_n)/2$ converges by showing it is Cauchy
negative_passages.docid: 120874
negative_passages.text: How to show if $|a_{n+1} - a_{n}| \le \frac{1}{2^n}$ then the sequence is Cauchy.
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: 145863
negative_passages.text: sequence of positive numbers satisfying $a_{n+1}=\frac{2}{a_n+a_{n-1}}$, prove it converges
negative_passages.docid: 152764
negative_passages.text: Let ($x_n$) be a monotone sequence and contain a convergent subsequence. Prove that ($x_n$) is convergent.
negative_passages.docid: 111353
negative_passages.text: Show that $\{a_n\}$ defined by $a_{n+1}=\frac{a_n+2}{a_n+1}$ converges
negative_passages.docid: 141484
negative_passages.text: Convergence of a sequence by convergence of sub-subsequence
negative_passages.docid: 3111
negative_passages.text: How to prove that if $\lim_{n \rightarrow \infty}a_n=A$, then $\lim_{n \rightarrow \infty}\frac{a_1+...+a_n}{n}=A$
negative_passages.docid: 8715
negative_passages.text: Let $(a_n)_n$ be a convergent sequence of integers , what can we say about $(a_n)_n$
negative_passages.docid: 97047
negative_passages.text: Limit for sequence $a_{m+n}\leq a_m+a_n$
negative_passages.docid: 123253
negative_passages.text: Limit of a sequence $a_1=1;a_{n+1}=(n+1)(1+a_n)$
negative_passages.docid: 139971
negative_passages.text: If $\sum{a_n}$ converges does that imply that $\sum{\frac{a_n}{n}}$ converges?
negative_passages.docid: 99933
negative_passages.text: Determine whether $(x_n)$ converges if $x_1= a , x_{n+1} = x_n+\frac{1}{x_n}$ for $n \in \mathbb{N}$