Find the limit of $c_n$.

Question

Let $a_n$ be a sequence such that $\lim_{n\to\infty}a_n=a^{*}$. Let $c_n = max(a_n,a_{n+1}+a_{n+2}) $
Question: determine whether $c_n$ converges, and if it converges, find the limit.
What I had so far:
$\forall\epsilon>0\exists n_0>0\forall n\geq n_0 : |a_n-a^*|$ So it also holds:
$|a_{n+1}-a^*|<\frac{\epsilon}{2}$ and $|a_{n+2}-a^*|<\frac{\epsilon}{2}$. So
$|a_{n+1}-a^*+a_{n+2}-a^*|\leq |a_{n+1}-a^*|+|a_{n+2}-a^*|<\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon$. So
$|a_{n+1}+a_{n+2}-2a^*|<\epsilon$ so the limit of $c_n$ is $2a^*$.