It's very well known that if $X$ is an irreducible projective variety (feel free to assume that the base-field is $\mathbb{C}$), then any two points $x,y\in X$ can be connected by the image of a non-singular curve. An outline of the proof can be found here. Anyway, the stardard reference for the proof is *Abelian Varieties-Mumford*, Lemma at pg.56. I'm wondering if this result can be adapted in such a way to give a characterization of convergent sequences in $X$. More precisely, motivated also by this question, I would prove a result like this:

**(RS) Rough Statement:** Let $(x_n)\subset X$ be a sequence of points in $X$. Then $x_n\to x\in X$ if and only if there exists a non-singular curve $C$ together with a morphism $f\colon C\longrightarrow X$ and sequence $(c_n)\subset C$ converging to $c\in C$ such that $x_n=f(c_n)$ for any $n$ and $f(c)=x$.

**Rough Idea of the Proof:** Any two points $x_n$ and $x_{n+1}$ can be connected by the image of a map $f_n\colon C_n\longrightarrow X$, where $C_n$ is a non-singular curve. Then the first thing coming in mind is taking $C'_n:=\cup_{i=1}^nC_i$ and considering the closure (in the Zariski topology) of the limit $$C:=\overline{\lim_n C'_n }.$$

Of course issues could arise in this construction, for example $C$ can no lo longer be a curve. Therefore I think that in order to check that $C$ is actually a curve (if it is then we get our non-singular curve by taking its normalization) we have to refine the construction of the curves $C_n$. It's not clear to me how we can proceed, and especially, if **(RS)** is true. Any constructive comment or answer is well accepted.