query: Show that $ \limsup x_n ≤ \limsup y_n$ and $\liminf x_n ≤ \liminf y_n$
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: 119804
negative_passages.text: If $\lim_{n\to \infty}a_n = a\in \mathbb{R}$ . Prove that $\limsup_{n\to \infty}a_n x_n=a\limsup_{n\to \infty}x_n$ .
negative_passages.docid: 50160
negative_passages.text: The quotient norm on $\ell^{\infty}(\mathbb{N}) / c_0 (\mathbb{N})$ is given by $\limsup_{n \in \mathbb{N}} |x_n|$.
negative_passages.docid: 37489
negative_passages.text: Show that $ \lim_{(x,y) \to 0} \frac {|x|^ \alpha |y|^ \beta} {|x|^ \gamma + |y|^ \delta} \text {exists} \iff \alpha/\gamma + \beta/\delta > 1.$
negative_passages.docid: 13561
negative_passages.text: Show: $E(|X|)\leqslant\liminf_{n\to\infty} E(|X_n|)$
negative_passages.docid: 50019
negative_passages.text: Prove $0$ is a partial limit of $a_n$
negative_passages.docid: 80440
negative_passages.text: Limit of $(x_n)$ with $0<x_1<1$ and $x_{n + 1} = x_n - x_n^{n + 1}$
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: 150995
negative_passages.text: Show that $\liminf(\sqrt{x_n})=\sqrt{\liminf x_n}$
negative_passages.docid: 131927
negative_passages.text: lim inf $|a_n|=0 \implies \sum_{k=1}^\infty a_{n_k}$ converges
negative_passages.docid: 13696
negative_passages.text: Proving $\limsup\frac 1 {a_n}=\frac 1 {\liminf a_n}$ and $\limsup a_n\cdot \limsup \frac 1 {a_n} \ge 1$
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: 135700
negative_passages.text: Show that $\lim_{n \to \infty} (1/n)\sum_{i=1}^n f(x_i)$ exists
negative_passages.docid: 134051
negative_passages.text: Prove $x_n$ converges IFF $x_n$ is bounded and has at most one limit point
negative_passages.docid: 125011
negative_passages.text: Show lim sup (an) = lim inf (an) for increasing for increasing sequence of subsets.
negative_passages.docid: 2232
negative_passages.text: A question on limsup
negative_passages.docid: 59162
negative_passages.text: Prove $\liminf\limits_{n\to\infty} F(x_n)\le F(x)$