Table of Contents
What is limit superior of a sequence?
The superior limit is the larger of the two, and the inferior limit is the smaller of the two. The inferior and superior limits agree if and only if the sequence is convergent (i.e., when there is a single limit).
Does a convergent sequence always have a limit?
Hence for all convergent sequences the limit is unique. Notation Suppose {an}n∈N is convergent. Then by Theorem 3.1 the limit is unique and so we can write it as l, say.
How do you prove Lim sup?
Proof. From Theorem 1.1 we know that lim inf sn = min(S) ≤ max(S) = lim sup sn. Now let us prove the equivalence between convergence and equality of liminf with limsup. If the sequence is convergent to L, then we know that any subsequence can only converge to L.
What is the difference between supremum and limit superior?
If the set of all limit points of a sequence is bounded above, then its supremum is said to be the limit superior (lim sup) of the sequence. Otherwise, and this happens precisely when the sequence itself is not bounded above, the lim sup of the sequence is taken as +infinity.
How do you prove a limit is bounded?
A function f : A → R is bounded on B ⊂ A if there exists M ≥ 0 such that |f(x)| ≤ M for every x ∈ B. A function is bounded if it is bounded on its domain.
How do you determine if a sequence is bounded above or below?
A sequence is bounded if it is bounded above and below, that is to say, if there is a number, k, less than or equal to all the terms of sequence and another number, K’, greater than or equal to all the terms of the sequence. Therefore, all the terms in the sequence are between k and K’.
How do I limit superior in latex?
The limit superior is usually denotes as \limsup . The \lim and \sup are there because these are also concepts on their own.
What are limit points and convergent limit points?
LIMIT POINTS AND CONVERGENCE De\fnition. A numberais said a limit point of (xn) if there exists a subsequence of (xn)convergent toa. Proposition.If (xn) is convergent, then any subsequence is convergent to the same limit. In other words,if(xn)converges tox, thenxis theonlylimit point of(xn).
What is the limit point of (xn)?
De\fnition. A numberais said a limit point of (xn) if there exists a subsequence of (xn)convergent toa. Proposition.If (xn) is convergent, then any subsequence is convergent to the same limit. In other words,if(xn)converges tox, thenxis theonlylimit point of(xn).
How do you find the finite bound of a set?
To get this bound, we take the supremum of | s | + 1 and all terms of | a n | when n ≤ N. Since the set we’re taking the supremum of is finite, we’re guaranteed to have a finite bound M.