query: Is the sequence $( d(x_{n},y_{n}))$ convergent if $X = \mathbb{R}$ with standard topology.
positive_passages.docid: 19599
positive_passages.text: $ d(x_n,y_n) \to d(x,y)$ if $x_n\to x$ and $y_n \to y$
negative_passages.docid: 84818
negative_passages.text: Convergence of $\sum_{n=1}^{\infty}x_n\overline{y_n}$ for every $(x_n)\in c_0$ implies $\sum_{n=1}^{\infty}|y_n|<\infty$
negative_passages.docid: 49507
negative_passages.text: Is this sequence convergent?
negative_passages.docid: 60282
negative_passages.text: Prove that $f_n$ converges uniformly on $\mathbb{R}$
negative_passages.docid: 17231
negative_passages.text: How I can prove that the sequence $\sqrt{2} , \sqrt{2\sqrt{2}}, \sqrt{2\sqrt{2\sqrt{2}}}$ converges to 2?
negative_passages.docid: 30167
negative_passages.text: Convergence of $\sqrt[n]{x^n+y^n}$ (for $x, y > 0$)
negative_passages.docid: 61085
negative_passages.text: $(P_n)$ converges to p if and only if every subsequence of $(P_n)$ converges to p
negative_passages.docid: 132713
negative_passages.text: The space $c_0$ of sequences that converge to 0 is not reflexive
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: 10933
negative_passages.text: Let $(x_n)$ be a bounded but not convergent sequence. Prove that $(x_n)$ has two subsequences converging to different limits.
negative_passages.docid: 47740
negative_passages.text: Let $x_1=a>0$ and $x_{n+1}=x_n+\frac{1}{x_n} \forall n\in \mathbb N$. Check whether the following sequence converges or diverges.
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: 3404
negative_passages.text: What are the convergent sequences in the cofinite topology
negative_passages.docid: 1760
negative_passages.text: Limit of the geometric sequence $\{r^n\}$, with $|r| < 1$, is $0$?
negative_passages.docid: 140379
negative_passages.text: Cauchy sequence in a discrete space
negative_passages.docid: 153121
negative_passages.text: show that $\{x_n\}$ is a cauchy sequence
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.