# Trying to prove if $S$ is a subset of $R$, every adherent point to $S$ is the limit of a sequence in $S$

I know the definition of an adherent point to a set $S$, a point $a$ is adherent to $S$ is given $\epsilon$ there exists am $x\in S$ such that $|x-a|<\epsilon$.

However I am having trouble understanding the proof for the following statement: if $S$ is a subset of $R$, every adherent point to $S$ is the limit of a sequence in $S$.

In particular it is the first part of the proof that I do not get, it says:

If $v$ is adherent to $S$ then given $n$ we can find $x_n \in S$ such that $|x_n -v|<1/n$. But I do not see how this follows from the fact that $v$ is adherent to $S$.

First of all, the standard term for adherent point is limit point. Now, we suppose that $v$ is a limit (adherent) point of $S$. Let $\epsilon=\frac 1n$, since $v$ is a limit point of $S$ there exist a point $x_n\in S$ such that $|x_n-v|<\epsilon$, therefor $|x_n-v|<\frac 1n$.