accumulation point and limit point

[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?
Please help me with that. Thanks in advance

[fantispug]

Welcome to PF!

It looks like you're on the right to me, though the question is pretty lousy. What have you got to doubt about your argument?

If it helps an equivalent formulation in a Hausdorff space (so in the real/complex numbers) is "L in a space X is an accumulation point of S iff every open set containing L contains a point in S other than L". What does this say about your example?