ozkan12 said:Let $\left\{{x}_{n}\right\}$ be a sequence...İf $\left\{{x}_{2n}\right\}$ is caucy sequence, can we say that $\left\{{x}_{n}\right\}$ is cauchy sequence ?
ozkan12 said:Hi Sudharaka,
İn some fixed point theorem, to prove that $\left\{{x}_{n}\right\}$ is cauchy sequence, author show that $\left\{{x}_{2n}\right\}$ is cauchy sequence...And in fixed point theorems, we use iteration sequence such that ${x}_{n}=f{x}_{n-1}$...İf we construct $\left\{{x}_{n}\right\}$ in this way, can we say that $\left\{{x}_{n}\right\}$ is cauchy sequence by proving that $\left\{{x}_{2n}\right\}$ is cauchy sequence ?
ozkan12 said:Sudharaka, I learned this information, Thank you for your attention, best wishes...:)
A Cauchy sequence is a sequence of numbers in which the terms become arbitrarily close to each other as the sequence progresses. In other words, for any given tolerance level, there exists a point in the sequence after which all subsequent terms are within that tolerance level of each other.
A Cauchy sequence and a convergent sequence both have the property that the terms get closer and closer together. However, a convergent sequence has a specific limit towards which the terms converge, while a Cauchy sequence does not necessarily have a limit and may not converge at all.
A subsequence is a sequence that is formed by selecting some terms from another sequence in the same order. These selected terms do not need to be consecutive in the original sequence, but their relative order must be maintained.
Yes, it is possible for a subsequence of a Cauchy sequence to be non-Cauchy. This can happen if the terms in the subsequence do not get arbitrarily close to each other as the sequence progresses, even though the terms in the original sequence do.
A metric space is said to be complete if every Cauchy sequence in that space converges to a limit within that space. In other words, completeness is a property that guarantees the convergence of all Cauchy sequences within a given metric space.