$\displaystyle \{a_n\}_{n=1}^{\infty}\to A$ and define $\displaystyle b_n=\frac{a_n+a_{n+1}}{2}, \ \ \forall n$

Prove $\displaystyle \{b_n\}_{n=1}^{\infty}\to A$

Is it as simple as $\displaystyle b_n=\frac{A+A}{2}=A$

Therefore, $\displaystyle \{b_n\}_{n=1}^{\infty}\to A$