Basic Properties of the Limit Inferior and Limit Superior of a Bounded Real Sequence

Let $(a_{n})_{n\in\mathbb{N}}$ and $(b_{n})_{n\in\mathbb{N}}$ be bounded sequences of real numbers, with limit inferior and limit superior as in those definitions, and let $a$ and $M$ be real numbers. Then the following hold.

1. (Order.) $\liminf_{n}a_{n}\le\limsup_{n}a_{n}$.

2. (Comparison.) If $a_{n}\le b_{n}$ for every $n\in\mathbb{N}$, then $\liminf_{n}a_{n}\le\liminf_{n}b_{n}$ and $\limsup_{n}a_{n}\le\limsup_{n}b_{n}$. In particular, if $a_{n}\le M$ for every $n$, then $\limsup_{n}a_{n}\le M$; and if $M\le a_{n}$ for every $n$, then $M\le\liminf_{n}a_{n}$.

3. (Eventual bounds.) For every real $\varepsilon>0$ there is $N\in\mathbb{N}$ such that $\liminf_{n}a_{n}-\varepsilon<a_{m}$ for every $m\ge N$, and there is $N'\in\mathbb{N}$ such that $a_{m}<\limsup_{n}a_{n}+\varepsilon$ for every $m\ge N'$.

4. (Extraction along the limit inferior.) For every real $\varepsilon>0$ there are natural numbers $n_{1}<n_{2}<n_{3}<\dots$ such that $a_{n_{j}}<\liminf_{n}a_{n}+\varepsilon$ for every $j\in\mathbb{N}$.

5. (Convergence criterion.) The sequence $(a_{n})_{n\in\mathbb{N}}$ has limit $a$ if and only if $\liminf_{n}a_{n}=a$ and $\limsup_{n}a_{n}=a$.