query: Prove $a_n$ and $b_n$ have the same limit
positive_passages.docid: 17289
positive_passages.text: prove $\sqrt{a_n b_n}$ and $\frac{1}{2}(a_n+b_n)$ have same limit
negative_passages.docid: 5181
negative_passages.text: $(z_{2n})_{n\in \mathbb{N}}$ and $(z_{2n+1})_{n\in \mathbb{N}}$ converges to same limit. Show $(z_n)_{n\in \mathbb{N}}$ converges to the same limit.
negative_passages.docid: 73934
negative_passages.text: Subadditive sequences and the limit $\sup_{n\to\infty} a_n / n$
negative_passages.docid: 34025
negative_passages.text: Limit of the Sequence: $a_{n+1}=a_n+\frac{1}{a_n}$ with $a_1=1$
negative_passages.docid: 112388
negative_passages.text: Suppose $\lim \limits_{n \to ∞} a_n=L$. Prove that $\lim\limits_{n \to ∞} \frac{a_1+a_2+\cdots+a_n}{n}=L$
negative_passages.docid: 68700
negative_passages.text: Proving Two Limits Are Equal to Each Other
negative_passages.docid: 7815
negative_passages.text: Prove that $\lim_{n\to\infty}a_n^2=a^2$ when ${a_n}$ is a sequence in $\mathbb{R}$ with $\lim_{n\to\infty}a_n=a$
negative_passages.docid: 116811
negative_passages.text: Proofs with limit superior and limit inferior: $\liminf a_n \leq \limsup a_n$
negative_passages.docid: 113798
negative_passages.text: Let $b_n$ decrease monotonically to zero, prove $\sum b_nz^n$ converges for $|z|\leq 1$ and $z\neq 1$
negative_passages.docid: 48500
negative_passages.text: Proof of convergence of sequences
negative_passages.docid: 149287
negative_passages.text: Prove that $A,B>0$ and $A_{ij}0$
negative_passages.docid: 35447
negative_passages.text: If $\sum a_n$ converges and $b_n=\sum\limits_{k=n}^{\infty}a_k $, prove that $\sum \frac{a_n}{b_n}$ diverges
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 $\lim_\limits{n \to \infty} a_n = \infty$.
negative_passages.docid: 64109
negative_passages.text: Let $a = \liminf x_n$, $b = \limsup x_n$. If $\lim (x_{n+1} - x_n) = 0$ then every element of $(a,b)$ is a subsequential limit of $(x_n)$
negative_passages.docid: 108929
negative_passages.text: Proving the limit of a function of a sequence is equal to the function of the limit of that sequence
negative_passages.docid: 47668
negative_passages.text: Show that $lim_{n \rightarrow \infty}m(O_n)=m(E)$ when $E$ is compact.
negative_passages.docid: 104339
negative_passages.text: Prove that all but finite number of members of sequence are positive if limit is positive