Intereting Posts

Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$?
What's the probability of rolling at least $k$ on $n$ dice with $s_1,\ldots,s_n$ sides?
suggest textbook on calculus
Show that $\mathfrak{S}=\bigcup_{N=1}^{\infty}\mathfrak{Z}_N\cup\left\{\emptyset\right\}$ is a semi-ring
When should one learn about $(\infty,1)$-categories?
How is the epsilon-delta definition of continuity equivalent to the following statement?
$f \in K$ of degree $n$ with Galois group $S_n$, why are there no non-trivial intermediate fields of $K \subset K(a)$ with $a$ a root of $f$?
If $ f(f(f(x)))=x$, does$ f(x)=x$ necessarily follow?
Find a sequence of r.v's satisfying the following conditions
Why does Group Theory not come in here?
Eigenvalues in terms of trace and determinant for matrices larger than 2 X 2
Definite integral, quotient of logarithm and polynomial: $I(\lambda)=\int_0^{\infty}\frac{\ln ^2x}{x^2+\lambda x+\lambda ^2}\text{d}x$
What is the average of rolling two dice and only taking the value of the higher dice roll?
Covariance of product of two functions of two binomial distributions
Where are the geometric figures in those “advanced geometry” textbooks?

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.

- The set of rationals has the same cardinality as the set of integers
- Prove that $A \subset B$ if and only if $A \setminus B = \emptyset$
- inverse element in a field of sets
- Is the axiom of choice really all that important?
- How to solve probability when sample space is infinite?
- Different arrows in set theory: $\rightarrow$ and $\mapsto$
- Relative sizes of sets of integers and rationals revisited - how do I make sense of this?
- The set of all functions from $\mathbb{N} \to \{0, 1\}$ is uncountable?
- $A ⊂ B$ if and only if $A − B = ∅$
- Borel Measures: Atoms (Summary)

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 boundandgreatest lower boundof a sequence of subset does mean ?

See here :

In mathematics, the

infimumof 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 termgreatest lower boundis also commonly used.The definition of

greatest lower boundseasily 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 boundof 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, orleast 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 limitis a subset (i.e. a lower bound) forall but finitely manyelements [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).

- Calculating the limit of the derivative of a sum of digamma functions
- Is $\sqrt1+\sqrt2+\dots+\sqrt n$ ever an integer?
- Game: two pots with coins
- Gateaux and Frechet derivatives on vector valued functions
- A few clarifications on the irrational slope topology
- What does it mean when a function is finite?
- Is there an elegant bijective proof of $\binom{15}{5}=\binom{14}{6}$?
- How to prove convergence of polynomials in $e$ (Euler's number)
- Elementary number theory: sums of primes and squares
- Transfinite series: Uncountable sums
- Relationship of aspect ratio to the homography matrices between a rectangle and an arbitrary quadrilateral
- the logarithm of quaternion
- Find the residue at $z=-2$ for $g(z) = \frac{\psi(-z)}{(z+1)(z+2)^3}$
- finding $\lim\limits_{n\to\infty }\dfrac{1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}}{1+\frac{1}{3}+\frac{1}{5}+\cdots+\frac{1}{2n+1}}$
- Proving $n^{97}\equiv n\text{ mod }4501770$