query: Positive, real, bounded sequence $(X_n)$. Prove that $\liminf \sqrt{X_n} = \sqrt{\liminf X_n}$
positive_passages.docid: 150995
positive_passages.text: Show that $\liminf(\sqrt{x_n})=\sqrt{\liminf x_n}$
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: 52712
negative_passages.text: Sequence converges iff $\limsup = \liminf$
negative_passages.docid: 66441
negative_passages.text: Prove that $|x|<1$ implies that $\lim_{n \to \infty} x^n=0$
negative_passages.docid: 106960
negative_passages.text: Proof to sequences in real analysis
negative_passages.docid: 136384
negative_passages.text: Inequality concerning limsup and liminf of Cesaro mean of a sequence
negative_passages.docid: 120648
negative_passages.text: Proof Check: Every Cauchy Sequence is Bounded
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: 29836
negative_passages.text: Prove: Convergent sequences are bounded
negative_passages.docid: 130877
negative_passages.text: Boundedness and Cauchy Sequence: Is a bounded sequence such that $\lim(a_{n+1}-a_n)=0$ necessarily Cauchy?
negative_passages.docid: 22470
negative_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: 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: 101630
negative_passages.text: Limit $\lim_{n\to\infty}\sqrt n(\sqrt[n]x-1)$
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: 26837
negative_passages.text: Limit of the sequence $\{n^n/n!\}$, is this sequence bounded, convergent and eventually monotonic?
negative_passages.docid: 11586
negative_passages.text: Sequences of Real Numbers in a closed bounded interval