Example 1 Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{1}=a, a_{2}=b, 2 a_{n+2}=a_{n+1}+a_{n} \text {. }
$$
(1) If $b_{n}=a_{n+1}-a_{n}$, prove: for $a \neq b$, $\left\{b_{n}\right\}$ is a geometric sequence;
(2) If $\lim _{n \rightarrow \infty}\left(a_{1}+a_{2}+\cdots+a_{n}\right)=4$, find the values of $a$ and $b$.
(2011, Joint Autonomous Admission Examination of Excellent Talent Cultivation Cooperative Universities)