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\}.