Proving Equivalence between conditions on sequences

Let E be a set of real numbers. Then p is an accumulation point of E iff the following condition holds:

there is a sequence (x_n) of members of E, each different from p, such that (xn) converges to p.

I have to prove that the above condition is equivalent to:

there is a sequence (x_n) of DISTINCT members of E converging to p.

At first I thought I just had to show the second condition with E-{p}. I'm not sure how to prove equivalence between these two conditions. If someone could help me out that'd be great!