Upper and Lower Limits of a Bounded Sequence

The greatest and smallest limit points of a bounded sequence, as given by the preceding tutorial, are respectively called the upper (or superior) and lower (or inferior) limits of the sequence.

The upper limit of a bounded sequence $u$ is denoted by $\overline {\mathop {\lim }\limits_{n \to \infty } } {\text{ }}{u_n}$ or $\overline {\lim } {\text{ }}u$. Similarly, the lower limit is denoted by $\mathop {\underline {\lim } }\limits_{x \to \infty } {\text{ }}{u_n}$ or $\underline {{\text{lim}}} {\text{ }}u$. Evidently, $\overline {{\text{lim}}} {\text{ }}{u_n} \geqslant \underline {{\text{lim}}} {\text{ }}{u_n}$.

For bounded ${u_n}$, the limits $\overline {{\text{lim}}} {\text{ }}{u_n},\underline {{\text{lim}}} {\text{ }}{u_n}$ shall also be defined as

$\overline {{\text{lim}}} {\text{ }}{u_n} = \lim {\text{ }}{a_n}$
$\underline {{\text{lim}}} {\text{ }}{u_n} = \lim {\text{ }}{{\text{b}}_n}$

where $\left\langle {{a_n}} \right\rangle ,\left\langle {{b_n}} \right\rangle$ are defined by ${a_n} = \sup \left\{ {{u_r}:r \geqslant n} \right\}$, ${b_n} = \inf \left\{ {{u_r}:r \geqslant n} \right\}$.