I'll state the question first.
In any metric space $(X,d)$, assume that $(x_n)$ is a sequence such that $x_n \to x$ for some $x \in X$. If $(y_m)$ is a sequence in $\{x_n: n \in \mathbb{N}\} \cup \{x\}$ such that $y_m \to x$ then is it true that $(y_m)$ must eventually be a subsequence of $(x_n)$?
My definition for an "eventually subsequence" is : there is a map $k:\mathbb{N} \to \mathbb{N}$ such that for all $m \in \mathbb{N}$ there is $n \in \mathbb{N}$ such that $y_m=x_{k(n)}$ and $k$ has the property that there should exist $M \in \mathbb{N}$ such that for all $n_1,n_2 \geq M$ if $n_1
I guess this is intuitively a reasonable claim. But I can't prove it rigorously. Any help (counter-examples, proofs etc.) will be appreciated. Thanks and regards.