Suppose that $\displaystyle (x_n)$ is a sequence of real numbers. Define a sequence $\displaystyle (y_n)$ by $\displaystyle y_n=\frac{x_n+x_{n+1}}{2}$ $\displaystyle \forall n \in \mathbb{N}$

(a) Prove that $\displaystyle (y_n)$ converges to a real number $\displaystyle x$ if $\displaystyle (x_n)$ converges to $\displaystyle x$.

(b) If $\displaystyle (y_n)$ converges, does $\displaystyle (x_n)$ also converge?