compact set and convergent sequence

Using the definition of a compact set: "A subset K of R is compact if every open cover of K has a finite subcover of K, that is if {Oj}j in A is an open cover of K, then there exists j1, j2, j3.... , jn in A s.t. K is a subset of the Union (from i=1 to n) of Oji."

I need to show that having a convergent sequence {Pn} in R, with lim = p, the set B = {p} U {pn: n in N} is a compact subset of R.

Proof:

Since {Pn} is a convergent sequence, it is nonempty. So for every e>0, the collection of of Ne(x):x in Pn} is an open cover of B. For every open cover, there is a finite subcover of B. Also, B is a subset of Union of open covers of B.

I don't think my proof is clear at all. Please provide some hints.