Suppose $(X,d)$ be a metric space. Let $(a_n)$ be a sequence in $X$ such that $(a_{2n-1})$ and $(a_{2n})$ has no Cauchy subsequence. Is it also true that $(a_n)$ has no Cauchy subsequence?
Let $A=\{a_{2n}:n\in\mathbb{N}\}$, is it true every Cauchy sequence in $A$ is constant?
True : If $b_n$ is a subsequence in $\{ a_n\}$ s.t. it is Cauchy, then one of them $\{ a_{2n}\},\ \{ a_{2n-1}\}$ contains $b_n$ infinitely many. If $ \{ a_{2n}\}$ contains a subsequence $\{ c_n\}$ of $\{ b_n\}$ (Note that $c_n$ is a Cauchy), then contradiction.