query: A trivial question on the sequences $\{a_n\}$ and $\{a_{n+1}\}$ converging to the same limit using the definition of convergence only
positive_passages.docid: 25206
positive_passages.text: How to prove that $\lim\limits_{n\to \infty}a_{n+1}=\lim\limits_{n\to \infty}a_{n}$
negative_passages.docid: 153736
negative_passages.text: Question about convergence of a simple recursive sequence
negative_passages.docid: 32777
negative_passages.text: Prove: If a sequence converges, then every subsequence converges to the same limit.
negative_passages.docid: 19657
negative_passages.text: Does the recursive sequence $a_1 = 1, a_n = a_{n-1}+\frac{1}{a_{n-1}}$ converge?
negative_passages.docid: 131927
negative_passages.text: lim inf $|a_n|=0 \implies \sum_{k=1}^\infty a_{n_k}$ converges
negative_passages.docid: 93545
negative_passages.text: Showing the sequence converges $a_{1}=\frac{1}{2}$, $a_{n+1}=\frac{1}{2+a_{n}}$
negative_passages.docid: 33976
negative_passages.text: Prove that $\lim_{n \to \infty} \sqrt[n]{a} = 1$ using only the basic properties of sequences
negative_passages.docid: 18675
negative_passages.text: $\{a_n\}$ sequence $a_1=\sqrt{6}$ for $n \geq 1$ and $a_{n+1}=\sqrt{6+a_n}$ show it that convergence and also find $\lim_{x \to \infty} \{a_n\}$
negative_passages.docid: 12031
negative_passages.text: Asymptotic behavior of the sequence $a_n=\sqrt{n+a_{n-1}}$
negative_passages.docid: 17779
negative_passages.text: Proving the Limit of a Sequence of without the Dominated Convergence Theorem
negative_passages.docid: 155695
negative_passages.text: The convergence of a sequence
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: 26837
negative_passages.text: Limit of the sequence $\{n^n/n!\}$, is this sequence bounded, convergent and eventually monotonic?
negative_passages.docid: 142509
negative_passages.text: If $\alpha\in (0,1]$ and $x_n=\alpha x_{n-1}+(1-\alpha)x_{n-2}$, show that the sequence $\{x_n\}$ is convergent.
negative_passages.docid: 60217
negative_passages.text: Proof: Series converges $\implies $ the limit of the sequence is zero
negative_passages.docid: 121774
negative_passages.text: Prove that $ \left(a_{n}\right)_{n=1}^{\infty} $ converges when $|a_{n+1}-a_{n}|<q|a_{n}-a_{n-1}|$ for $ 0<q<1 $
negative_passages.docid: 53001
negative_passages.text: If $|f'(c)| \leq M$ and $ M < 1$, the sequence defined by $a_{n+1} = f(a_n)$ converges