Show that $a_{n+1} = \frac{1}{2}(a_n+b_n), $ $b_{n+1} = \frac{1}{2}(b_n+c_n) $and $c_{n+1} = \frac{1}{2}(c_n+a_n)$ are convergent, 
Question: Let $a_0,b_0,c_0$ be real numbers.
Define the sequences $(a_n)_n,(b_n)_n, (c_n)_n$ recursively by
$$a_{n+1} = \frac{1}{2}(a_n+b_n), \quad b_{n+1} = \frac{1}{2}(b_n+c_n) \quad \text{and}\quad c_{n+1} = \frac{1}{2}(c_n+a_n).$$
Prove that the sequences are convergent and find their limits.

My attempt: I try to find the closed-form solution for $a_n,b_n,c_n$.
Note that
$$
\begin{pmatrix}
a_{n+1} \\
b_{n+1} \\
c_{n+1} \\
\end{pmatrix} = \begin{pmatrix}
\frac{1}{2} & \frac{1}{2} & 0 \\
0 & \frac{1}{2} & \frac{1}{2} \\
\frac{1}{2} & 0 & \frac{1}{2}
\end{pmatrix} \begin{pmatrix}
a_n \\
b_n \\
c_n
\end{pmatrix} = A\begin{pmatrix}
a_n \\
b_n \\
c_n
\end{pmatrix}.
$$
Clearly
$$
\begin{pmatrix}
a_{n} \\
b_{n} \\
c_{n} \\
\end{pmatrix} = A^{n}\begin{pmatrix}
a_0 \\
b_0 \\
c_0 \\
\end{pmatrix}.
$$
To compute $A^n$, I try to diagonalize $A$ over complex numbers.
From Wolfram alpha, $A$ has eigenvalues $1, \frac{1}{4}(1+i\sqrt3)$ and $\frac{1}{4}(1-i\sqrt3)$.
Therefore,
$$
\begin{pmatrix}
a_{n} \\
b_{n} \\
c_{n} \\
\end{pmatrix} = X \begin{pmatrix}
1 & 0 & 0 \\
0 & \left( \frac14 (1+i\sqrt 3) \right)^n & 0 \\
0 & 0 & \left( \frac14 (1-i\sqrt 3) \right)^n
\end{pmatrix} X^{-1} \begin{pmatrix}
a_0 \\
b_0 \\
c_0 \\
\end{pmatrix}
$$
for some invertible matrix $X$ (in fact, columns of $X$ are eigenvectors of $A$).
However, I stuck at here.
I do not know how to proceed to show that the sequences $(a_n), (b_n), (c_n)$ convergent.
It would be good if someone can solve this problem by continuing my method above.

EDIT: After reading comments below, I attempted the following.
From Wolfram alpha, we have
$$\lim_{n\to\infty} A^n = \frac13 J_3$$
where $J_3$ is the $3\times 3$ matrix with all entries $1$.
Therefore,
$$\lim_{n\to\infty} \begin{pmatrix}
a_{n} \\
b_{n} \\
c_{n} \\
\end{pmatrix} = \frac13 J_3 \begin{pmatrix}
a_{0} \\
b_{0} \\
c_{0} \\
\end{pmatrix} = \frac13 \begin{pmatrix}
a_{0} + b_{0} + c_{0} \\
a_{0} + b_{0} + c_{0} \\
a_{0} + b_{0} + c_{0}
\end{pmatrix}.$$
Hence, the sequences $(a_n),(b_),(c_n)$ are  convergent and their limits are $\frac13 (a_0 + b_0 + c_0)$.

Hint: $$a_{n+1}+b_{n+1}+c_{n+1}=a_n+b_n+c_n=\ldots=a_0+b_0+c_0$$
$$|a_{n+1}-b_{n+1}|+|b_{n+1}-c_{n+1}|+|c_{n+1}-a_{n+1}|=\frac12\,(|a_n-b_n|+|b_n-c_n|+|c_n-a_n|)$$
