What is a monotonic subsequence?
Definition 14 A subsequence of a sequence xn is a sequence yn such that there exists a function. f : N → N strictly increasing such that yi = xf(i) ∀ i ∈ N. It turns out that every sequence of real numbers has subsequence that is monotone. Lemma 6 Every sequence of real numbers has a monotone subsequence.
What is monotone subsequence example?
Monotonicity: The sequence sn is said to be increasing if sn sn+1 for all n 1, i.e., s1 s2 s3 …. A sequence is said to be monotone if it is either increasing or decreasing. Example. The sequence n2 : 1, 4, 9, 16, 25, 36, 49, is increasing.
What is the statement of monotone subsequence theorem?
Informally, the theorems state that if a sequence is increasing and bounded above by a supremum, then the sequence will converge to the supremum; in the same way, if a sequence is decreasing and is bounded below by an infimum, it will converge to the infimum.
Is every convergent sequence monotone?
Theorem 2.4: Every convergent sequence is a bounded sequence, that is the set {xn : n ∈ N} is bounded. Sequences which are either increasing or decreasing are called monotone. The following result is an application of the least upper bound property of the real number system.
What is a subsequence in math?
In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements.
What does monotonic series mean?
It means that the sequence is always either increasing or decreasing, it the terms of the sequence are getting either bigger or smaller all the time, for all values bigger than or smaller than a certain value.
What do you mean by limit of a sequence?
The limit of a sequence is the value the sequence approaches as the number of terms goes to infinity. Not every sequence has this behavior: those that do are called convergent, while those that don’t are called divergent. Limits capture the long-term behavior of a sequence and are thus very useful in bounding them.
Are all Cauchy sequences monotone?
If a sequence (an) is Cauchy, then it is bounded. Our proof of Step 2 will rely on the following result: Theorem (Monotone Subsequence Theorem). Every sequence has a monotone subsequence. If a subsequence of a Cauchy sequence converges to x, then the sequence itself converges to x.
Do all bounded sequences converge?
Every bounded sequence is NOT necessarily convergent.
How do you prove Heine Borel theorem?
Proof
- If a set is compact, then it must be closed.
- If a set is compact, then it is bounded.
- A closed subset of a compact set is compact.
- If a set is closed and bounded, then it is compact.
Can a sequence have many convergent subsequences?
Remark Notice that a bounded sequence may have many convergent subsequences (for example, a sequence consisting of a counting of the rationals has subsequences converging to everyreal number) or rather few (for example a convergent sequence has all its subsequences having the same limit). Proof Suppose the sequence (a1, a2, a3, a4,
Which is the best example of a subsequence?
A subsequenceis an infinite ordered subset of a sequence. Examples. (a2, a4, a6, ) is a subsequence of (a1, a2, a3, a4, So is (a1, a10, a100, a1000, Theorem. Any subsequence of a convergent sequence is convergent (to the same limit). Proof.
Which is a non decreasing number in the sequence?
More formally: Let N be a natural number which is greater than all the “nice” points. We define n 1 = N and n i + 1 := min { n > n i: a n ≥ a n i }. Hence ( a n i) is non-decreasing.
Who was the mathematician who discovered the subsequence?
The nicest thing about these subsequences is a result attributed to the Czech mathematician and philosopher Bernard Bolzano(1781 to 1848) and the German mathematician Karl Weierstrass(1815 to 1897). The Bolzano-Weierstrass Theorem