query: $d(x_{n+2},x_{n+1})≤rd(x_{n+1},x_n)$ gives convergent sequence
positive_passages.docid: 147661
positive_passages.text: Show that {$x_n$} is a convergent sequence if $d(x_{n+2}, x_{n+1})≤ rd(x_{n+1}, x_n)$
negative_passages.docid: 85217
negative_passages.text: Boundedness and convergence of $x_{n+1} = x_n ^2-x_n +1$
negative_passages.docid: 78886
negative_passages.text: Convergence of the sequence $\sqrt{1+2\sqrt{1}},\sqrt{1+2\sqrt{1+3\sqrt{1}}},\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1}}}},\cdots$
negative_passages.docid: 47922
negative_passages.text: Prove that the sequence $(a_n)$ defined by $a_0 = 1$, $a_{n+1} = 1 + \frac 1{a_n}$ is convergent in $\mathbb{R}$
negative_passages.docid: 148302
negative_passages.text: In convergence, why not $d \leq \epsilon$?
negative_passages.docid: 40797
negative_passages.text: $(a_n+a_{n+1})$ is convergent implies $(a_n/n)$ converges to $0$
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: 20136
negative_passages.text: Convergence of the sequence $ \sqrt {n-2\sqrt n} - \sqrt n $
negative_passages.docid: 100580
negative_passages.text: Proof that a sequence converges.
negative_passages.docid: 17779
negative_passages.text: Proving the Limit of a Sequence of without the Dominated Convergence Theorem
negative_passages.docid: 65592
negative_passages.text: Convergence of $a_n=(1/2)^{(1/3)^{...^{(1/n)}}}$
negative_passages.docid: 116414
negative_passages.text: Proving that the difference of the terms of a sequence, multiplied by $n$ converge
negative_passages.docid: 9044
negative_passages.text: Convergence of the sequence $(1+\frac{1}{n})(1+\frac{2}{n})\cdots(1+\frac{n}{n})$
negative_passages.docid: 82883
negative_passages.text: Show that the sequence $a_{n+1}=a_{n}+\dfrac{a^2_{n}}{n^2}$ is upper bounded
negative_passages.docid: 57095
negative_passages.text: How do I prove the sequence $x_{n+1}=\sqrt{\alpha +x_n}$, with $x_0=\sqrt \alpha$ converges? ( boundedness?)
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: 144786
negative_passages.text: Show that the following inductively defined sequence converges and find its limit.