Intuition of upper & lower bound of sequence of subsets.

I have come across these while studying the limsup & liminf of sequence of subset of a set. In order to understand that, I have to understand what least upper bound & greatest lower bound of a sequence of subset mean. I would be grateful if anyone helps me comprehend this concept intuitively as I am new & novice to this topic.

Solutions Collecting From Web of "Intuition of upper & lower bound of sequence of subsets."

Ref also to your subsequent question.

A partially ordered set is a very “simple” mathematical structure :

A pooset [Partially Ordered SET] consists of a set together with a binary relation that indicates that, for certain pairs of elements in the set, one of the elements precedes the other. Such a relation is called a partial order to reflect the fact that not every pair of elements need be related: for some pairs, it may be that neither element precedes the other in the poset.

An easy example is the set of human males with the “order relation” : “$x$ is father of $y$”.

Obviously, not two males $a$ and $b$ whatever are in the relation : “$a$ is father of $b$” or “$b$ is father of $a$”, but some are; this means that the order is partial.

Usually, the order relation is symbolized with $<$, because it is a “generalization” of the “less than” relation between numbers [that, by the way, is a partial order which is total].

Consider now the set $\mathcal P(X)$ of subsets of a set $X$; $\mathcal P(X)$ is partially orderes by the “inclusion” relation $\subseteq$.

Cosnider $X = \{a,b \}$; we have that the set $\mathcal P(X)$ containing all its subsets is $\{ \emptyset, \{a \}, \{ b \}, \{ a, b \} \}$.

As you can easily check, $\mathcal P(X)$ is partially ordered by $\subseteq$ :

$\emptyset \subseteq \{a \}, \{ b \}, \{ a, b \}$

$\{a \}, \{ b \} \subseteq \{ a, b \}$

but $\{a \} \nsubseteq \{ b \}$ and $\{b \} \nsubseteq \{ a \}$.


Thus :

what least upper bound and greatest lower bound of a sequence of subset does mean ?

See here :

In mathematics, the infimum of a subset $S$ of a partially ordered set $T$ is the greatest element of $T$ that is less than or equal to all elements of $S$. Consequently the term greatest lower bound is also commonly used.

The definition of greatest lower bounds easily generalizes to subsets of arbitrary partially ordered sets and as such plays a vital role in order theory.

The dual concept of infimum is given by the notion of a supremum or least upper bound.

The least upper bound of a subset $S$ of $(\mathcal P(X), \subseteq)$, where $\mathcal P(X)$ is the power set of some set $X$, is the supremum with respect to [the relation of inclusion] $\subseteq$, and is the union of the elements of $S$.


Regarding Limsup of a sequence $\{ X_n \}$ of subset of the set $X$ (trivial example : $X=[0,1]$ and $X_n=[0,1/n]$ ) :

consider the infimum, or greatest lower bound, of a sequence of sets. In the case of a sequence of sets, the sequence constituents “meet” at a set that is somehow smaller than each constituent set. Set inclusion provides an ordering that allows set intersection to generate a greatest lower bound $\bigcap X_n$ of sets in the sequence $\{ X_n \}$. Similarly, the supremum, or least upper bound, of a sequence of sets is the union $\bigcup X_n$ of sets in the sequence $\{ X_n \}$.

Thus :

If $\{ X_n \}$ is a sequence of subsets of $X$ [i.e. $X_n \subseteq X$, for every $n$], then:

$\text {lim sup} \ X_n$ consists of elements of $X$ which belong to $X_n$ for infinitely many $n$. That is, $x \in \text {lim sup} X_n$ if and only if there exists a subsequence $\{ X_{n_k} \}$ of $\{ X_n \}$ such that $x \in X_{n_k}$ for all $k$.

$\text {lim inf} \ X_n$ consists of elements of $X$ which belong to $X_n$ for all but finitely many $n$. That is, $x \in \text {lim inf} X_n$ if and only if there exists some $m > 0$ such that $x \in X_n$ for all $n > m$.

So the inferior limit acts like a version of the standard infimum that is unaffected by the set of elements that occur only finitely many times. That is, the infimum limit is a subset (i.e. a lower bound) for all but finitely many elements [of the sequence $\{ X_n \}$ ].

$x$ is in the lim sup of a sequence of sets if and only if it is in infinitely many of the sets.

And $x$ is in the lim inf of a sequence of sets if and only if it is in all but finitely many of the sets (or equivalently if it is in all the sets from some point on).