Is lim sup always greater than lim inf?

Is lim sup always greater than lim inf?

If the terms in the sequence are real numbers, the limit superior and limit inferior always exist, as the real numbers together with ±∞ (i. e. the extended real number line) are complete. The limit superior and limit inferior of a sequence are a special case of those of a function (see below).

How do you calculate lim sup and lim inf?

αn = sup {(−1)n(n + 5)/n, (−1)n+1(n + 6)/(n + 1),…} = (n + 5)/n for n even, and(n + 6)/(n + 1) for n odd → 1 as n → ∞. Therefore lim sup an = 1. Similarly lim inf an = −1.

What is the meaning of lim sup?

Lim Sup and Lim Inf. Informally, for a sequence in R, the limit superior, or lim sup, of a sequence is the largest subsequential limit. This implies that bt > M for every t, hence M is a lower bound for B, hence, by the least upper bound property (which implies the greatest lower bound property), inf B = lim supxt ∈ R.

Can lim sup infinite?

lim sup and lim inf always exist (possibly infinite) for any sequence of real numbers.

Why does Lim sup always exist?

The limit of a bounded sequence need not exist, but the liminf and limsup of a bounded sequence always exist as real numbers. When there’s no loss of clarity, we might omit the subscript variable (above, it is n). There are also shorter notations meaning the same thing: liman means lim supan and liman means lim inf a.

Can the infimum be infinity?

The infimum and supremum are the best possible lower and upper bounds of a set. They need not be real numbers; they can be ±∞ for unbounded sets.

Can the Infimum be infinity?

What does inf mean in math?

In mathematics, the infimum (abbreviated inf; plural infima) of a subset of a partially ordered set is a greatest element in that is less than or equal to all elements of if such an element exists. Consequently, the term greatest lower bound (abbreviated as GLB) is also commonly used.

Why does lim sup always exist?

How do you find sup and inf/of a sequence?

If M ∈ R is an upper bound of A such that M ≤ M′ for every upper bound M′ of A, then M is called the supremum of A, denoted M = sup A. If m ∈ R is a lower bound of A such that m ≥ m′ for every lower bound m′ of A, then m is called the or infimum of A, denoted m = inf A.

Is Infinity An upper bound?

The exact definition of infinity is contextual, but in general ∞ represents something increasing without bound (which means there is no upper bound for the function), and −∞ represents something decreasing without bound (which means there is no lower bound for the function).