Table of Contents
Is every Cauchy sequence has a convergent subsequence?
Every Cauchy sequence of real numbers is bounded, hence by Bolzano–Weierstrass has a convergent subsequence, hence is itself convergent. This proof of the completeness of the real numbers implicitly makes use of the least upper bound axiom.
How do you prove a Cauchy sequence is convergent?
The proof is essentially the same as the corresponding result for convergent sequences. Any convergent sequence is a Cauchy sequence. If (an)→ α then given ε > 0 choose N so that if n > N we have |an- α| < ε. Then if m, n > N we have |am- an| = |(am- α) – (am- α)| ≤ |am- α| + |am- α| < 2ε.
Is a sequence convergent if it has a convergent subsequence?
Theorem 3.4 If a sequence converges then all subsequences converge and all convergent subsequences converge to the same limit.
Is a subsequence of a Cauchy sequence Cauchy?
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.
How do you prove 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.
Can a convergent sequence have a divergent subsequence?
Accepted Answers: If every subsequence of a sequence converges then the sequence converges If a sequence has a divergent subsequence then the sequence itself is divergent. If a sequence is bounded and divergent then there are two subsequences that converge to different limits.
What is the difference between convergent and Cauchy sequence?
A Cauchy sequence is a sequence where the terms of the sequence get arbitrarily close to each other after a while. A convergent sequence is a sequence where the terms get arbitrarily close to a specific point. Formally a convergent sequence {xn}n converging to x satisfies: ∀ε>0,∃N>0,n>N⇒|xn−x|<ε.