Let $(a_{n})_{n\ge 0}$ and $(b_{n})_{n\ge 0}$ be two sequences with arbitrary real values $a_0, a_1, b_0, b_1$. For $n\ge 1$, let $a_{n+1}, b_{n+1}$ be defined in this way: $$a_{n+1}=\dfrac{b_{n-1}+b_{n}}{2}, b_{n+1}=\dfrac{a_{n-1}+a_{n}}{2}$$ Prove that for any constant $c>0$ there exists a positive integer $N$ s.t. for all $n>N$, $|a_{n}-b_{n}|<c$.