Table of Contents
Can a sequence have no convergent subsequences?
(a) An unbounded sequence has no convergent subsequences. Let (an) be a sequence which contains a bounded subsequence (ank ). Since (ank ) is a bounded sequence, it has a convergent subsequence by the Bolzano-Weierstrass Theorem.
How do you prove a sequence is convergent or divergent?
If limn→∞an lim n → ∞ exists and is finite we say that the sequence is convergent. If limn→∞an lim n → ∞ doesn’t exist or is infinite we say the sequence diverges.
Do unbounded sequences always diverge?
Every unbounded sequence is divergent. The sequence is monotone increasing if for every Similarly, the sequence is called monotone decreasing if for every The sequence is called monotonic if it is either monotone increasing or monotone decreasing.
Can an unbounded sequence have a Cauchy subsequence?
The sequence is strictly increasing but unbounded, so every subsequence is unbounded, whence no subsequence can converge.
How do you show a sequence is a subsequence?
The easiest way to approach the theorem is to prove the logical converse: if an does not converge to a, then there is a subsequence with no subsubsequence that converges to a. Let an be a sequence, and let us assume an does not converge to a. Let N=0.
Is the sequence (- 1 N N convergent?
However, different sequences can diverge in different ways. The sequence (−1)n diverges because the terms alternate between 1 and −1, but do not approach one value as n→∞. On the other hand, the sequence 1+3n diverges because the terms 1+3n→∞ as n→∞.
How do you determine if a sequence is a subsequence?
Definition. A subsequence of a sequence {aj}j≥j0 in Rn, is a new sequence, denoted {akj}j, where {kj} is an increasing sequence of integers such that kj≥j0 for every j. Thus, the jth term akj of the subsequence is the kjth term of the original sequence.