Prove that the supremum of the set of the sequence is bigger or equal then the limit of the sequence

Question

The question: Let $a_n$ with $n=0, 1, 2,\dots$ be a sequence that converges to $a^*$. We define also $A=\{a_n | n=0, 1, 2, \dots\}$. Prove that $\sup A \ge a^*$.

I've tried to prove it by contradiction with the formal definition of the limit and the supremum, but without success.