The lim inf is $3$ and the lim sup is $5$. The inf of the whole sequence is $3-\frac12$. If you throw away the first term or the first two terms, the inf of what's left is $3-\frac14$. If you throw …

Geometrically, $\lim \inf$ is the leftmost (or smallest) accumulation point of a sequence, while $\lim \sup$ is the rightmost (or biggest) one. If you can prove that the one on the left is in fact …

Clearly $\lim \inf A_n \subseteq \lim \sup A_n$. A series converges implies all the people who can't get another job eventually swallow their pride and become regulars too: $\lim \inf A_n$ = …

I will write only about the first inequality $$\limsup\limits_{n \rightarrow \infty} s_n + \liminf\limits_{n \rightarrow \infty} t_n \leq \limsup\limits_{n \rightarrow \infty} (s_n + t_n).$$ (You wrote that you …

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Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their …

Therefore the sequence $\{a_n\}$ converges iff its $\lim \inf$ equals its $\lim \sup$ (and, as a corollary, the limit of the sequence always equals that of the infima and suprema). Please tell …

2017年12月25日 · I may first give an example : finding limit $$ \lim_{x \rightarrow \infty} \frac{1+x}{x} $$ When we use straightforward approach, we get $$ \frac{\infty+1}{\infty} = …

2016年2月28日 · I have the question: Show that for bounded sequences $(x_n)$ and $(y_n)$: $$\\liminf\\limits_{n \\rightarrow \\infty} (x_n + y_n) \\leq \\limsup\\limits_{n ...

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