Extracting distinct sequence Let $(x_n)$ be a non-constant sequence in $\mathbb{R}$ and $x_n\rightarrow p$ for some $p\in \mathbb{R}$. Can I always extract a subsequence $(x_{n_k})$ whose all elements are distinct? (of course $x_{n_k}\rightarrow p$). Thanks in advance.
By non-constant I mean $\forall a\in\mathbb{R}, x_k=a$ for finitely many $k\in \mathbb{N}$

Yes, if by non-constant you really mean that no tail of the sequence is constant, i.e., that it’s not eventually constant. If you could not do so, there would be some finite $F\subseteq\Bbb R$ such that $x_n\in F$ for each $n\in\Bbb N$. But then it must be the case that $p\in F$ and the sequence is eventually constant at $p$.
