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\}.