Properties of the Limit Superior and the Limit Inferior: Tail Formula, Order, Comparison, Bounds, Eventual and Frequent Inequalities, Subsequences, Negation and Convergence

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness, let $(a_{n})$ and $(b_{n})$ be bounded sequences in $\mathbb{R}$, with limit superior and limit inferior as in The Limit Superior and the Limit Inferior of a Bounded Sequence of Real Numbers, convergence as in Convergent Sequences of Real Numbers §converges and subsequences as in Subsequences §subsequence, and let $A,M\in\mathbb{R}$ and $K\in\mathbb{N}$.

¶ With $\overline{a}_{k}$ and $\underline{a}_{k}$ the tail supremum and the tail infimum of $(a_{n})$ as in Tail Suprema and Tail Infima of a Bounded Sequence of Real Numbers §tails, $\limsup_{n\to\infty}a_{n}=\inf\{\overline{a}_{k}:k\in\mathbb{N}\}$ and $\liminf_{n\to\infty}a_{n}=\sup\{\underline{a}_{k}:k\in\mathbb{N}\}$.

¶ $\liminf_{n\to\infty}a_{n}\le\limsup_{n\to\infty}a_{n}$.

¶ If $a_{n}\le b_{n}$ for every $n\ge K$, then $\liminf_{n\to\infty}a_{n}\le\liminf_{n\to\infty}b_{n}$ and $\limsup_{n\to\infty}a_{n}\le\limsup_{n\to\infty}b_{n}$.

¶ If $a_{n}\le M$ for every $n\ge K$, then $\limsup_{n\to\infty}a_{n}\le M$; if $M\le a_{n}$ for every $n\ge K$, then $M\le\liminf_{n\to\infty}a_{n}$.

¶ For every real $\varepsilon>0$ there is $N\in\mathbb{N}$ with $\liminf_{n\to\infty}a_{n}-\varepsilon<a_{m}<\limsup_{n\to\infty}a_{n}+\varepsilon$ for every $m\in\mathbb{N}$ with $m\ge N$.

¶ For every real $\varepsilon>0$ and every $N\in\mathbb{N}$ there are $m,m'\in\mathbb{N}$ with $m\ge N$, $m'\ge N$, $\limsup_{n\to\infty}a_{n}-\varepsilon<a_{m}$ and $a_{m'}<\liminf_{n\to\infty}a_{n}+\varepsilon$.

¶ $(a_{n})$ has a subsequence converging to $\limsup_{n\to\infty}a_{n}$ and a subsequence converging to $\liminf_{n\to\infty}a_{n}$.

¶ If a subsequence of $(a_{n})$ converges to $L\in\mathbb{R}$, then $\liminf_{n\to\infty}a_{n}\le L\le\limsup_{n\to\infty}a_{n}$.

¶ For every real $\varepsilon>0$ there are strictly increasing maps $\sigma,\tau:\mathbb{N}\to\mathbb{N}$ with $\limsup_{n\to\infty}a_{n}-\varepsilon<a_{\sigma(j)}$ and $a_{\tau(j)}<\liminf_{n\to\infty}a_{n}+\varepsilon$ for every $j\in\mathbb{N}$.

¶ The sequence $(-a_{n})_{n\in\mathbb{N}}$ is bounded, $\limsup_{n\to\infty}(-a_{n})=-\liminf_{n\to\infty}a_{n}$ and $\liminf_{n\to\infty}(-a_{n})=-\limsup_{n\to\infty}a_{n}$.

¶ $a_{n}\to A$ if and only if $\liminf_{n\to\infty}a_{n}=A=\limsup_{n\to\infty}a_{n}$.