limsup and liminf

#RealAnalysis

The concept of $lim sup$ and $lim inf$ of a sequence , is to put a long term upper bound and lower bound of a sequence. The $lim sup$ of a sequence as: given the sequence $(y_{n})$ defined as follow:

y_{n} = sup {x_{k} ∣ k \geq n} and lim_{n \to \infty} y_{n} := lim sup x_{n} = a

For some $a$ , or as the following:

\forall ε > 0 \exists N \in N [\forall n \geq N (x_{n} - ε < a)]

And the $lim inf$ is defined similarly with a sequence $(z_{n})$ defined as:

z_{n} = inf {x_{k} ∣ k \geq n} and lim_{n \to \infty} z_{n} := lim inf x_{n} = b

for some $b$ , or as the following:

\forall ε > 0 \exists N \in N [\forall n \geq N (x_{n} < b + ε)]

The good thing is that $lim sup$ and $lim inf$ of a sequence exist with weaker conditions, as long as the sequence is bounded, both most exist, and even if it’s not bounded one of them can exist.

Algebraic Properties

Given a bounded sequence $(a_{n})$

lim inf a_{n} \leq lim sup a_{n}

The sequence $(a_{n})$ converges to a limit if and only if $lim inf a_{n} = lim sup a_{n}$

Given two sequences bounded $(a_{n})$ and $(b_{n})$ then, the following properties hold:

$lim sup (a_{n} + b_{n}) \leq lim sup a_{n} + lim sup b_{n}$
$lim inf (a_{n} + b_{n}) \geq lim inf a_{n} + lim inf b_{n}$
$lim sup (a_{n} b_{n}) \leq (lim sup a_{n}) (lim sup b_{n})$
$lim inf (a_{n} b_{n}) \geq (lim inf a_{n}) (lim inf b_{n})$
If a constant $c \in R^{+}$ then, $lim sup c a_{n} = c lim sup a_{n}$ and $lim inf c a_{n} = c lim inf a_{n}$
If a constant $c \in R^{-}$ then, $lim sup c a_{n} = c lim inf a_{n}$ and $lim inf c a_{n} = c lim sup a_{n}$