Accumulation point and limit point

1. Oct 25, 2009

thesleeper

1. The problem statement, all variables and given/known data
"If a sequence converges to L, then L is an accumulation point of {a_n|n greater than or equal to 1)."?
Prove or disprove the statement

2. Relevant equations
accumulation point is also a limit point

3. The attempt at a solution
I think the statement is not true. So in order to disprove it, I give an counterexample
consider the sequence {a_n} where a_n=L for all n. This sequence converges to 1, but its range is finite. Hence, this sequence has no accumulation point. Since the definition of an accumulation point of S is that every neighborhood of it contains infinitely many points of the S. Then, the statement is not always true.
Am I correct? and if the statement is true, how do you prove it?