query: If $\{a_n\}$ is a sequence such that $0\leq a_{m+n}\leq a_m+a_n$, show that $\lim_{n\to\infty}\frac{a_n}{n}$ exists
positive_passages.docid: 22470
positive_passages.text: Prove $\lim_{n\to\infty} \frac{a_n}{n}$ exists for positive sequence where $a_{n+m} \leq a_n + a_m$
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: 31930
negative_passages.text: Given the sequence $(x_n)$ is unbounded, show that there exist a subsequence $(x_{n_k})$ such that $\lim(1/x_n)=0$.
negative_passages.docid: 91009
negative_passages.text: Convergence of the series $\sum a_n$ implies the convergence of $\sum \frac{\sqrt a_n}{n}$, if $a_n>0$
negative_passages.docid: 135700
negative_passages.text: Show that $\lim_{n \to \infty} (1/n)\sum_{i=1}^n f(x_i)$ exists
negative_passages.docid: 84969
negative_passages.text: When does $\lim_{n\rightarrow \infty} a_n = 0$ imply $\sum_{n = 0}^{\infty} a_n$ converges?
negative_passages.docid: 60024
negative_passages.text: Proof that a sequence is convergent
negative_passages.docid: 12031
negative_passages.text: Asymptotic behavior of the sequence $a_n=\sqrt{n+a_{n-1}}$
negative_passages.docid: 27932
negative_passages.text: Show that $\lim\limits_{n\to\infty}\frac{a_1+a_2+\dots+a_n}{n^2}$ exists and is independent of the choice of $a$
negative_passages.docid: 118023
negative_passages.text: Let $(p_n)_{n \in \mathbb{N}}$ be the sequence of prime numbers, then $\lim_{n \to \infty}\frac{p_{n+1}}{p_n} = 1$?
negative_passages.docid: 155695
negative_passages.text: The convergence of a sequence
negative_passages.docid: 86319
negative_passages.text: Give an example of a bounded, non-convergent real sequence $(a_n)$ s.t. $a_n-a_{n-1}\rightarrow 0$
negative_passages.docid: 143227
negative_passages.text: Show that: $\lim \limits_{n\to\infty}\frac{x_n-x_{n-1}}{n}=0 $
negative_passages.docid: 76527
negative_passages.text: If $x_{m+n} \le x_n+x_m$, then $\lim x_n/n$ exists and is equal to $\inf x_n/n$
negative_passages.docid: 53688
negative_passages.text: How can one prove that $\lim_{n \to \infty}a^{1/n}=1$ for every $a>0$?
negative_passages.docid: 102582
negative_passages.text: If $(a_n)$ is a decreasing sequence of strictly positive numbers and if $\sum{a_n}$ is convergent, show that $\lim{na_n}=0$
negative_passages.docid: 19745
negative_passages.text: Let $a_n$ be a sequence such that: $ a_{n+1}-a_n \ge \frac{1}{n}$. prove that $\lim_\limits{n \to \infty} a_n = \infty$.