Limit inferior and limit superior

In mathematics, the limit inferior and limit superior (or limes inferior and limes superior) of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant.

Limit inferior is also called infimum limit, limit infimum, liminf, inferior limit, lower limit, or inner limit; limit superior is also known as supremum limit, limit supremum, limsup, superior limit, upper limit, or outer limit.

The limit inferior of a sequence

(

x

n

)

{\displaystyle (x_{n})}

is denoted by

lim inf

n

→

∞

x

n

or

lim

_

n

→

∞

⁡

x

n

,

{\displaystyle \liminf _{n\to \infty }x_{n}\quad {\text{or}}\quad \varliminf _{n\to \infty }x_{n},}

and the limit superior of a sequence

(

x

n

)

{\displaystyle (x_{n})}

is denoted by

lim sup

n

→

∞

x

n

or

lim

¯

n

→

∞

⁡

x

n

.

{\displaystyle \limsup _{n\to \infty }x_{n}\quad {\text{or}}\quad \varlimsup _{n\to \infty }x_{n}.}

Definition for sequences

The limit inferior of a sequence

(

x

n

)

{\displaystyle (x_{n})}

is defined by

lim inf

n

→

∞

x

n

:=

lim

n

→

∞

(

inf

m

≥

n

x

m

)

{\displaystyle \liminf _{n\to \infty }x_{n}:=\lim _{n\to \infty }\!{\Big (}\inf _{m\geq n}x_{m}{\Big )}}

or

lim inf

n

→

∞

x

n

:=

sup

n

≥

0

inf

m

≥

n

x

m

=

sup

{

inf

{

x

m

:

m

≥

n

}

:

n

≥

0

}

.

{\displaystyle \liminf _{n\to \infty }x_{n}:=\sup _{n\geq 0}\,\inf _{m\geq n}x_{m}=\sup \,\{\,\inf \,\{\,x_{m}:m\geq n\,\}:n\geq 0\,\}.}

Similarly, the limit superior of

(

x

n

)

{\displaystyle (x_{n})}

is defined by

lim sup

n

→

∞

x

n

:=

lim

n

→

∞

(

sup

m

≥

n

x

m

)

{\displaystyle \limsup _{n\to \infty }x_{n}:=\lim _{n\to \infty }\!{\Big (}\sup _{m\geq n}x_{m}{\Big )}}

or

lim sup

n

→

∞

x

n

:=

inf

n

≥

0

sup

m

≥

n

x

m

=

inf

{

sup

{

x

m

:

m

≥

n

}

:

n

≥

0

}

.

{\displaystyle \limsup _{n\to \infty }x_{n}:=\inf _{n\geq 0}\,\sup _{m\geq n}x_{m}=\inf \,\{\,\sup \,\{\,x_{m}:m\geq n\,\}:n\geq 0\,\}.}

Alternatively, the notations

lim

_

n

→

∞

⁡

x

n

:=

lim inf

n

→

∞

x

n

{\displaystyle \varliminf _{n\to \infty }x_{n}:=\liminf _{n\to \infty }x_{n}}

and

lim

¯

n

→

∞

⁡

x

n

:=

lim sup

n

→

∞

x

n

{\displaystyle \varlimsup _{n\to \infty }x_{n}:=\limsup _{n\to \infty }x_{n}}

are sometimes used.

The limits superior and inferior can equivalently be defined using the concept of subsequential limits of the sequence

(

x

n

)

{\displaystyle (x_{n})}

.