Does a convergent subsequence imply a bounded sequence?

The theorem states that each bounded sequence in Rn has a convergent subsequence. An equivalent formulation is that a subset of Rn is sequentially compact if and only if it is closed and bounded. The theorem is sometimes called the sequential compactness theorem.

Is it true that a bounded sequence which contains a convergent subsequence is convergent?

Proof: Every sequence in a closed and bounded subset is bounded, so it has a convergent subsequence, which converges to a point in the set, because the set is closed. Conversely, every bounded sequence is in a closed and bounded set, so it has a convergent subsequence.

Can an unbounded sequence have a convergent subsequence?

(a) An unbounded sequence has no convergent subsequences. Since (ank ) is a bounded sequence, it has a convergent subsequence by the Bolzano-Weierstrass Theorem. This convergent subsequence is a subsequence of the original sequence by problem 2. Thus the contrapositive of statement (b) is true.

Is every bounded sequence convergent?

Every bounded sequence is NOT necessarily convergent. Let an=sin(n).

What is a bounded subsequence?

If a sequence is bounded, then every its subsequence is bounded. 2. If a sequence converges to a (that could be +∞ or −∞), then every its subsequence also converges to a. Exercise 4.11. Prove that a monotone sequences which contains a bounded subsequence is bounded.

Can a subsequence be the original sequence?

No. A subsequence is a sequence taken from the original, where terms are selected in the order they appear.

Is every subsequence of an unbounded sequence unbounded?

One example is the sequence ( ) of natural numbers. The sequence is strictly increasing but unbounded, so every subsequence is unbounded, whence no subsequence can converge.

Is a sequence bounded?

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’.

Does every bounded sequence have a convergent subsequence?

The one mentioned above has two subsequences that converge, the one with only zeroes and the the one with only ones. The Bolzano–Weierstrass theorem states that every bounded sequence in has a convergent subsequence. Originally Answered: Does every bounded sequence convergent or every bounded sequance have a subsequence that converges?

Are infinite sequences always bounded by the domain?

They could be anything, and have just about any behaviour. Unless, of course, your domain only allows one value, in which case all infinite sequences converge, or the values in the domain are bounded, in which case all sequences are bounded, or… That depends on the subsequence. If you mean “just any old subsequence”, then no.

What is the difference between a Cauchy sequence and a bounded sequence?

In the second example, the sequence is Cauchy, but the metric space under consideration fails to be complete. However, a bounded sequence always has a Cauchy subsequence, and in Euclidean and other complete metric spaces, always has a convergent subsequence: Bolzano–Weierstrass theorem – Wikipedia.

Does the sequence converge to a given number?

The sequence itself, however, now certainly doesn’t converge to anything: it has an infinite subsequence that goes haywire. So now we have a necessary condition: the subsequences in the promised list cannot afford to miss, between them, infinitely many numbers.