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 away all the terms up to that one and the one after it, the inf of what's left is $3-\frac16$.

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 bigger than or equal to the one on the right, then they coincide.

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: $\lim \inf A_n$ = $\lim \sup A_n$.

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 can prove the rest.)

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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 me if you see any holes or things I could've done better, as I've made many stupid mistakes in proofs before. Thanks for your help!

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 $$ \frac{\infty+1}{\infty} = \frac{\infty}{\infty} $$ In the process of investigating a limit, we know that both the numerator and denominator are going to infinity.. but we dont know the behaviour of each dynamics.

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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