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 $S$.