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 $${a_n} = \sup \left\{ {{u_r}:r \geqslant n} \right\}$$, $${b_n} = \inf \left\{ {{u_r}:r \geqslant n} \right\}$$.