What does $\lim \inf _ { n \rightarrow \infty }$ mean? I'm new into Mathematical Analysis, and my textbook says:

Consider the series: $\frac { 1 } { 2 } + \frac { 1 } { 3 } + \frac {
 1 } { 2 ^ { 2 } } + \frac { 1 } { 3 ^ { 2 } } + \frac { 1 } { 2 ^ { 3
 } } + \frac { 1 } { 3 ^ { 3 } } + \frac { 1 } { 2 ^ { 4 } } + \frac {
 1 } { 3 ^ { 4 } } + \cdots$
for which 
$\liminf _ { n \rightarrow \infty } \frac { a _ { n + 1 } } { a _ { n
 } } = \lim _ { n \rightarrow \infty } \left( \frac { 2 } { 3 } \right)
 ^ { n } = 0$
$\liminf _ { n \rightarrow \infty } \sqrt [ n ] { a _ { n } } = \lim _
 { n \rightarrow \infty } \sqrt [ 2 n ] { \frac { 1 } { 3 ^ { n } } } =
 \frac { 1 } { \sqrt { 3 } }$
$\limsup _ { n \rightarrow \infty } \sqrt [ n ] { a _ { n } } = \lim _
 { n \rightarrow \infty } \sqrt [ 2 n ] { \frac { 1 } { 2 ^ { n } } } =
 \frac { 1 } { \sqrt { 2 } }$
$\limsup _ { n \rightarrow \infty } \frac { a _ { n + 1 } } { a _ { n
 } } = \lim _ { n \rightarrow \infty } \frac { 1 } { 2 } \left( \frac {
 3 } { 2 } \right) ^ { n } = + \infty$

My question is: What does $\lim \inf _ { n \rightarrow \infty }$ mean? 
And why $\lim \inf _ { n \rightarrow \infty } \frac { a _ { n + 1 } } { a _ { n > } } = \lim _ { n \rightarrow \infty } \left( \frac { 2 } { 3 } \right)$?

You can refer to your notes or to the wiki page Limit superior and limit inferior for the definition.
It is just a generalization for the limit of a sequence, in particular we consider
$$I_n=\inf\{a_n:k\ge n\}$$
and since $I_n$ is an increasing sequence the limit always exist (finite or infinite) and then we can consider and define
$$\liminf_{n\to \infty} a_n =\lim_{n\to \infty} I_n $$
To show that for a given sequence $a_n$ we have $\liminf_{n\to \infty} a_n =L$ we need to show two fact


*

*$a_n \ge b_n \quad b_n\to L$

*find a sub-sequence $a_{i_n}$ such that $a_{i_n}\to L$
