A sequence is bounded if it is bounded above and below, i.e. if there is a number k that is less than or equal to all the terms of the sequence and another number, K, that is greater than or equal to all the terms of the sequence. Therefore all terms of the sequence lie between k and K.
How to determine if a sequence is bounded or unbounded?
A sequence an is a bounded sequence if it is bounded above and below. If a sequence is not bounded, it is an unbounded sequence. For example, the sequence 1/n is bounded because 1/n≤1 for any positive integer n. It is also bounded from below because 1/n≥0 for all positive integers n.
How do you prove that a convergent sequence is bounded?
We will now examine an important theorem that says that a convergent sequence is also bounded. Theorem: If a convergent sequence , i.e. $\lim_{n \to \infty} a_n = L$ for some , then it is also bounded , i.e. for some , $\mid a_n \mid ≤ M$.
How do you find the upper bound of a sequence?
For A(n) you probably know (or can believe) that A(n)→3 holds as n→∞. So A(n)≤4 for large n. More specifically, write A(n)=3+4n−1n2−n+1 and find N such that 4n−1≤n2−n+1 for all n≥N is applicable. Then A(n)≤4 for all n≥N.
Is a bounded sequence always convergent?
Remark: It is true that every bounded sequence contains a convergent subsequence, and furthermore every monotone sequence converges if and only if it is bounded. See entry added to monotone convergence theorem for more information on guaranteed convergence of bounded monotone sequences.
How do you know if a function has an upper or lower bound?
As far as I know a bounded sequence can be either convergent or finitely oscillating, it can’t be divergent because as a bounded sequence it can’t diverge infinitely.
Are all bounded sequences Cauchy?
e) TRUE Every bounded sequence has a Cauchy subsequence. We have proved that every bounded sequence (sn) has a convergent subsequence (snk), but all convergent sequences are Cauchy, so (snk) is Cauchy.
Are all convergent sequences Cauchy?
Every convergent sequence is a Cauchy sequence. However, the reverse may not be true. For the sequences in Rk the two terms are the same. More generally, we call an abstract metric space X such that every Cauchy sequence in X converges to a point in X a complete metric space.