A Cauchy sequence is a sequence of numbers in which the terms gradually get closer together. Formally, a sequence (an) is Cauchy if for any positive real number ε, there exists a positive integer N such that for all m,n > N, |am - an| < ε.
A quasi-Cauchy sequence is a sequence of numbers in which the terms get arbitrarily close, but not necessarily converge. Formally, a sequence (an) is quasi-Cauchy if for any positive real number ε, there exists a positive integer N such that for all m,n > N, |am - an| < ε.
The main difference between a Cauchy sequence and a quasi-Cauchy sequence is that a Cauchy sequence is required to converge, while a quasi-Cauchy sequence may or may not converge. Additionally, a Cauchy sequence must have its terms getting closer together as n increases, while a quasi-Cauchy sequence only needs to have its terms getting arbitrarily close.
To prove that a subsequence of a Cauchy sequence is quasi-Cauchy, we can use the definition of a Cauchy sequence and a subsequence. Let (an) be a Cauchy sequence and (ank) be a subsequence of (an). Then, for any positive real number ε, there exists a positive integer N such that for all m,n > N, |am - an| < ε. Since (ank) is a subsequence, we can choose N to be the same as in (an), and the inequality will still hold for all m,n > N, proving that (ank) is quasi-Cauchy.
This statement is important because it establishes a relationship between two types of sequences - Cauchy and quasi-Cauchy. It also highlights the fact that a subsequence of a Cauchy sequence may not necessarily be Cauchy, but will always be quasi-Cauchy. Additionally, this result is useful in proving the convergence of sequences and series, and has applications in various fields such as analysis, topology, and number theory.