If a series is convergent and contains only irrational numbers, does the limit also have to be irrational?

Question

I have encountered this problem. If $(a_n)_{n \ge 1}$ is a sequence such that $a_n \in \mathbb{R} \setminus \mathbb{Q}$ for every natural number $n$ and the $l = \lim_{n \to \infty} a_n$ is a real number, does that mean that $l \in \mathbb{R} \setminus \mathbb{Q}$? The original question that I have found didn't mention the fact that $l$ is a number and such I was able to provide the series $a_n = (\sqrt {2} + \sqrt {3}) ^ n$ which contains only irrational numbers and the limit if $\infty$. But what if the series is convergent?