query: Prove that $\limsup\limits_{n \to \infty}{a_n} = \lim\limits_{N \to \infty} \sup{\{a_n: n \ge N\}}$
positive_passages.docid: 43197
positive_passages.text: Showing that two definitions of $\limsup$ are equivalent
negative_passages.docid: 117640
negative_passages.text: Prove that $\lim \inf \dfrac{x_{n+1}}{x_n}\leq \lim \inf x_n^{1\over n} \leq \lim \sup x_n^{1\over n} \leq \lim \sup \dfrac{x_{n+1}}{x_n}. $
negative_passages.docid: 44507
negative_passages.text: How can I find $\lim_{n\to \infty} a_n$
negative_passages.docid: 17779
negative_passages.text: Proving the Limit of a Sequence of without the Dominated Convergence Theorem
negative_passages.docid: 111205
negative_passages.text: Proving that $\lim_{k\to \infty}\frac{f(x_k)-f(y_k)}{x_k-y_k}=f'(c)$ where $\lim_{k\to \infty}x_k=\lim_{k\to \infty}y_k=c$ with $x_k<c<y_k$
negative_passages.docid: 81302
negative_passages.text: Showing that $a_n \in \ell^{\infty}$
negative_passages.docid: 74835
negative_passages.text: how to prove that $ \lim x_n^{y_n}=\lim x_n^{\lim y_n}$?
negative_passages.docid: 45988
negative_passages.text: limit of Holder norms: $\sup\limits_{x\in [a,b]} f(x) = \lim\limits_{n\rightarrow\infty} \left(\int_a^b (f(x))^n \;dx\right)^{\frac{1}{n}}$
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: 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: 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: 54952
negative_passages.text: Is it true that $\limsup\sqrt[n]{|a_n|}=\limsup\sqrt[n]{|a_n-a_{n-1}|}$, ($n>1$), where $\{a_n\}$ does not have a finite limit?
negative_passages.docid: 38860
negative_passages.text: $P(\limsup A_n)=1 $ if $\forall A \in \mathfrak{F}$ s.t. $\sum_{n=1}^{\infty} P(A \cap A_n) = \infty$
negative_passages.docid: 125011
negative_passages.text: Show lim sup (an) = lim inf (an) for increasing for increasing sequence of subsets.
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: 113141
negative_passages.text: Prove that $\lim_{n \to \infty}\bigg[\int_0^1f(t)^n \text{dt}\bigg]^{1/n}=M$
negative_passages.docid: 68609
negative_passages.text: Let $a_{n+1}=\sqrt{a_1+a_2+\cdots+a_n}$ .Prove that $ \lim\limits_{n \rightarrow \infty} \frac{a_n}{n}=\frac{1}{2}$