query: If $a_n$ and $b_n$ are converging sequences, and for all $n\in\mathbb Z+$, $a_n \leq b_n$, then $\lim_{n\to\infty} a_n \leq \lim_{n\to\infty} b_n$.
positive_passages.docid: 23140
positive_passages.text: Given $a_n, b_n$ are both convergent sequences such that $a_n \leq b_n$ for all $n$, show $\lim_{n\to\infty} a_n \leq \lim_{n\to\infty} b_n$.
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
negative_passages.docid: 1525
negative_passages.text: If for any $k$, $\sum\limits_{n=0}^\infty a_n^k=\sum\limits_{n=0}^\infty b_n^k$, then$(a_n)=(b_{σ(n)}),\ σ \in{\mathfrak S}_{\mathbb N}$
negative_passages.docid: 19225
negative_passages.text: Let $\{a_n\}$ be defined as follows: $a_1 = 2$, $a_{n+1}=\frac{1}{3-a_n}$, if $n \geq 1$. Does $\{a_n\}$ converge?
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: 102523
negative_passages.text: Show that $\lim\limits_{n \to \infty} \sup\limits_{k \geq n} \left(\frac{1+a_{k+1}}{a_k}\right)^k \ge e$ for any positive sequence $\{a_n\}$
negative_passages.docid: 131425
negative_passages.text: Prove that the convergence of $\prod_{n=1}^{\infty} b_n$ implies $\lim_{n→∞} b_n = 1$
negative_passages.docid: 73823
negative_passages.text: $\lim_{n\to\infty} a_n=a$ if and only if $\forall p\in \Bbb N$, $\lim_{n\to\infty} |a_{n+p}-a_n|=0$
negative_passages.docid: 133886
negative_passages.text: Existence of increasing sequence $\{x_n\}\subset S$ with $\lim_{n\to\infty}x_n=\sup S$
negative_passages.docid: 88064
negative_passages.text: Prove: $a_n \leq b_n \implies \limsup a_n \leq \limsup b_n$
negative_passages.docid: 112663
negative_passages.text: Let $a_n$ be a sequence s.t $a_1 > 0 \land a_{n+1} = a_n + \dfrac{1}{a_n}$. Prove that $a_n$ is increasing and tends to infinity
negative_passages.docid: 155475
negative_passages.text: $\lim\limits_{n\to \infty} \int_0^1 |f(x)-a_nx-b_n| dx=0$ implies $(a_n)_n,(b_n)_n$ are convergent
negative_passages.docid: 89325
negative_passages.text: $a_{m+n}\leq a_ma_n$ implies the convergence of $\sqrt[n]a_n$
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: 54659
negative_passages.text: How prove that $(a_{n})_{n \in \mathbb{N}}$ is convergent with $a_{0}=1$, $a_{n+1}= \sqrt{2a_{n}}$?
negative_passages.docid: 115411
negative_passages.text: if $A_n \longrightarrow \infty $ and $B_n \longrightarrow \infty $ then $(A_n+B_n) \longrightarrow \infty$
negative_passages.docid: 124615
negative_passages.text: Proving convergence of sums: If $\sum_{n=1}^{\infty}a^2_n$ and $\sum_{n=1}^{\infty}b^2_n$ converge then $\sum_{n=1}^{\infty}a_nb_n$ converges