query: If $a_n \geq b_n$ and $b_n$ converges to $b$ then $\liminf a_n\geq b$
positive_passages.docid: 102478
positive_passages.text: $X_n\leq Y_n$ implies $\liminf X_n \leq \liminf Y_n$ and $\limsup X_n \leq \limsup Y_n$
negative_passages.docid: 149068
negative_passages.text: Prove: Assume $\lim_{n\to\infty}(a_n)=a, \lim_{n\to\infty}(b_n)=b$ then $\lim_{n\to\infty}(a_n+b_n)=a+b$
negative_passages.docid: 150995
negative_passages.text: Show that $\liminf(\sqrt{x_n})=\sqrt{\liminf x_n}$
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: 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: 30488
negative_passages.text: Convergence of a sequence implies convergence of the norm of the sequence
negative_passages.docid: 144786
negative_passages.text: Show that the following inductively defined sequence converges and find its limit.
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: 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: 111353
negative_passages.text: Show that $\{a_n\}$ defined by $a_{n+1}=\frac{a_n+2}{a_n+1}$ converges
negative_passages.docid: 94066
negative_passages.text: Prove that: {$f_n$} converges unifromly to $f$ on $E$ as $n \rightarrow \infty$ if and only if $\lim_{n\rightarrow \infty} ||f_n-f||_u = 0$
negative_passages.docid: 114975
negative_passages.text: Limit of sequences $V(n)$ and $U(n)$
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: 154578
negative_passages.text: Proving there is a sequence convergent to a limit point of a set without axiom of countable choice?
negative_passages.docid: 837
negative_passages.text: Suppose that $(s_n)$ converges to $s$, $(t_n)$ converges to $t$, and $s_n \leq t_n \: \forall \: n$. Prove that $s \leq t$.
negative_passages.docid: 76137
negative_passages.text: Too simple proof for convergence of $\sum_n a_n b_n$?
negative_passages.docid: 152525
negative_passages.text: If $b_n$ is a bounded sequence and $\lim a_n = 0$, show that $\lim(a_nb_n) = 0$