Proving equivalence of two definitions of "dense" in $\mathbb R$ Ok, so I've been trying to learn how to write proofs and I've been given this definition for DENSE:
D is dense in $R$ if for each x $\in$ $R$, $\exists$ {${d_n}$}$_{n=1}{^\infty}$ $\in$ D with $\lim_{n \to \infty} d_n$ = x
How would I go about starting to prove this? Is there a particular method that would be easiest?
EDIT: This is another definition I was given. 
A set D $\subseteq R$ is DENSE if for each a < b, a,b $\in R$, $\exists$ d $\in$ D with a < d < b
 A: The bottom definition is, in English,

A subset $D$ of $\mathbb{R}$ is dense if every open interval contains a member of $D$.

and the definition you are trying to prove is

A subset $D$ of $\mathbb{R}$ is dense if every point in $\mathbb{R}$ is the limit of some sequence of elements of $D$.

To prove that these are equivalent, you will have to show that the first implies the second and that the second implies the first.
To show that the first implies the second, begin by assuming that we have a set $D$ satisfying the first definition, that is every open interval in $\mathbb{R}$ contains a member of $D$.  Using this information, can you construct some sequence $\{d_n\}$ that converges to a limit $x$?  Perhaps by using a sequence of intervals containing or approaching $x$?
To show that the first implies the second, begin by assuming that every point $x \in \mathbb{R}$ is a limit of some sequence $\{d_n\}$.  If $d_n \rightarrow x$, what can we say about the existence of some $d_i$ in an open interval containing $x$?
