How do you find the limit of a supremum?

We define lim supx→∞f(x) to be the maximum limit of any sequence f(xn) where xn→∞. More precisely, define S:={(xn):xn→∞}, and L:={limf(xn):(xn)∈S, and limf(xn) exists }. Then lim supx→∞f(x):=maxL.

Is a supremum a limit?

It is not generally true that the supremum of a set A in R is a limit point of that set. For example, as pointed out in the comments by Daniel Fischer, for any a∈R, we have sup{a}=a, but a is not a limit point of {a}.

What is meant by limit inferior and limit superior?

Definition for a set The limit inferior of a set X ⊆ Y is the infimum of all of the limit points of the set. That is, Similarly, the limit superior of a set X is the supremum of all of the limit points of the set.

What is the Supremum of a sequence?

The supremum (abbreviated sup; plural suprema) of a subset of a partially ordered set is the least element in that is greater than or equal to all elements of if such an element exists. Consequently, the supremum is also referred to as the least upper bound (or LUB).

Is the infimum a limit point?

The limit superior of a bounded sequence is its largest limit point and its limit infimum is its smallest limit point.

What is the difference between cluster point and limit point?

In analysis and topology, the open balls around a limit point contain at least one element of the set, and the open balls around a cluster point contain infinitely many elements of the set.

What is meant by limit superior?

The limit superior of is the smallest real number such that, for any positive real number , there exists a natural number such that for all . In other words, any number larger than the limit superior is an eventual upper bound for the sequence. Only a finite number of elements of the sequence are greater than .

What is the limit supremum of a point?

The limit supremum ( which is also called limit superior or Upper Limit ) of at the point is an extended real number ( ie real number or ) which is denoted by Suppose is a very small arbitrary positive number such that For each such we get is an extended real number. We can easily verify that decreases as decreases.……………………………. (1)

What is the limit infimum?

The limit infimum ( which is also called limit inferior or Lower Limit ) of [math]\\;\\;f\\;\\;[/math] at the point [math]\\;\\;c\\;\\; [/math]is an extended real number which is denoted by [math]\\;\\;\\displaystyle \\lim_{\\substack x\o c} inf\\;f(x)\\;.\\;[/math]

Can the supremum be the maximum of an infinite series?

In infinite series, the supremum is sometimes the maximum. Example – for all reals from 0 to 1, 1 is both the supremum and the maximum. However, for many infinite sets, no maximum exists, therefore the supremum cannot be the maximum.

What is the limit inferior of at the point?

The limit infimum ( which is also called limit inferior or Lower Limit ) of at the point is an extended real number which is denoted by For each such we get is an extended real number. We can easily verify that increases as decreases.……………………………. (2) Therefore exists as an extended real number (say) is called the limit inferior of at the point