Infinitely many positive terms in a Conditionally Convergent Series

I am trying to prove the following:
If $$\Sigma_{n=0}^{\infty}a_n$$ is conditionally convergent but not absolutely convergent, then there are infinitely many positive terms.

My approach is to assume that there are finitely many positive terms, then conclude that the series must be absolutely convergent.
However, I am not sure how to formalize this.
Could you give some guidelines for proving this?
Thanks.

If there are finitely many positive terms, let's assume the last one is $$a_k$$. Then we can split the series into $$\sum_{n=0}^k a_n$$ + $$\sum_{n=k+1}^\infty a_n$$. Can you show that this second sum, which only has negative terms, converges absolutely?