lim inf (x_n) = inf {x: infinitely many x_n are < x }
i cant understand what they are saying here?
i was told to see lim inf as the infimum of all the limits of all the subsequences
they say something else
??
Yeah I am not quite sure what that means. I have always been taught that if we have a sequence $\displaystyle \left\{a_n\right\}$ and a set $\displaystyle S$ which is the set of all the subsequential limits of $\displaystyle \left\{a_n\right\}$. Then $\displaystyle \liminf a_n=\inf S$. This is exactly the same thing you said.
Hello,
You can view this way :
$\displaystyle \liminf A_n=\lim_{n \to \infty} \bigcap_{k>n} A_k=\bigcup_n \bigcap_{k >n} A_k$
$\displaystyle x \in \liminf A_n \Leftrightarrow \exists n \in \mathbb{N}, ~ \forall k >n,~ x \in A_k$
This basically means that x belongs to $\displaystyle A_n$ for infinitely many indices, except a finite number of them.
This is similar to the definition for a valued sequence $\displaystyle (x_n)$
$\displaystyle \liminf x_n=\lim_{n \to \infty} \uparrow \inf_{k>n} x_k$
This means that $\displaystyle \inf_{k>n} x_k<x$
The standard definition of lim inf is $\displaystyle L = \liminf\{x_n\} = \sup_n\inf_{k>n}x_k$. But the sequence $\displaystyle \{\inf_{k>n}x_k\}$ increases with n, so its sup is equal to its limit: $\displaystyle L = \lim_{n\to\infty}\inf_{k>n}x_k$. That of course is why it is called lim inf.
An equivalent definition is $\displaystyle \liminf\{x_n\} = \inf \{x: \text{infinitely many }x_n \text{ are }< x \}$. To see that this is equivalent to the standard definition, you need to show that if there are only finitely many x_n with x_n<x, then x≤L; and if there are infinitely many x_n<x, then x≥L.
A third possible definition is that the lim inf is the infimum of all the limits of all the convergent subsequences of the sequence $\displaystyle (x_n)$. This definition is helpful because it is easier to visualise than the standard definition. It is also equivalent to the standard definition. But you have to remember to interpret convergence as including convergence to $\displaystyle \pm\infty$, because a sequence might not have any subsequences that converge to finite limits.
so when your are saying
"that we take sum X
for which there are infinitely many members of Xn sequence which are smaler than X"
that the definition of an upper bound
lim inf is a infimum of a group of limits of all sub sequences
i cant see how both of them are the same thing
??
and why say "infinite number of members of Xn"
we could say "all the members of Xn"