query: Proof: $\exists z\in \mathbb{C}$ such that every sub-sequence of $(z_n)_n$ has a convergent sub-sequence with the limit $z$.
positive_passages.docid: 17201
positive_passages.text: $y_{2n}, y_{2n+1}$ and $y_{3n}$ all converge. What can we say about the sequence $ y_n$?
negative_passages.docid: 87011
negative_passages.text: Prove that for the series $\sum_{n \in \mathbb{N}}|\zeta_n\mu_n|$ to be convergent for all $\zeta \in l^p \implies \mu \in l^q$
negative_passages.docid: 13724
negative_passages.text: What can be said about the convergence on $|z|=1$?
negative_passages.docid: 76137
negative_passages.text: Too simple proof for convergence of $\sum_n a_n b_n$?
negative_passages.docid: 55614
negative_passages.text: Prove that a set consisting of a sequence and its limit point is closed
negative_passages.docid: 48593
negative_passages.text: Proof that if $s_n \leq t_n$ for $n \geq N$, then $\liminf_{n \rightarrow \infty} s_n \leq \liminf_{n \rightarrow \infty} t_n$
negative_passages.docid: 76593
negative_passages.text: Show that $f_n(z)=z^n$ converges locally uniformly on $|z|<1$ but not uniformly(?)
negative_passages.docid: 46830
negative_passages.text: Prove that the sequence $a_{n+1} =\frac{1}{2}\left(a_{n}+\frac{c}{a_{n}}\right)$ is convergent and find its limit
negative_passages.docid: 83040
negative_passages.text: Proving that the sequence $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$ is boundedly convergent on $\mathbb{R}$
negative_passages.docid: 143545
negative_passages.text: If $(a_n)$ is Cauchy it has a subsequence $(a_{n_i})$ such that $d(a_{n_{i+1}},a_{n_i})<2^{-i}$ for all $i$.
negative_passages.docid: 153121
negative_passages.text: show that $\{x_n\}$ is a cauchy sequence
negative_passages.docid: 23140
negative_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: 46323
negative_passages.text: Criteria for convergent sequence (Baby Rudin Theorem 3.22)
negative_passages.docid: 17779
negative_passages.text: Proving the Limit of a Sequence of without the Dominated Convergence Theorem
negative_passages.docid: 53005
negative_passages.text: Properties of $\liminf$ and $\limsup$ of sum of sequences: $\limsup s_n + \liminf t_n \leq \limsup (s_n + t_n) \leq \limsup s_n + \limsup t_n$
negative_passages.docid: 142057
negative_passages.text: Prove that if $\sum_{n=1}^\infty a_n$ converges then $\sum_{k=1}^\infty A_k$ converges
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.