Definition
lim inf
\liminf
and
lim sup
\limsup
and
lim
\lim
lim inf
A
n
=
∪
n
=
1
∞
∩
k
=
n
∞
A
k
\liminf A_n = \cup_{n=1}^{\infty} \cap_{k=n}^{\infty} A_k
lim sup
A
n
=
∩
n
=
1
∞
∪
k
=
n
∞
A
k
\limsup A_n = \cap_{n=1}^{\infty} \cup_{k=n}^{\infty} A_k
lim
A
n
=
lim sup
A
n
\lim A_n = \limsup A_n
if
lim sup
A
n
=
lim inf
A
n
\limsup A_n = \liminf A_n