Real sequence/Arithmetical mean/Dependent on start value/Convergence/Exercise

Let ${}a\in \mathbb {R}$. For initial value ${}x_{0}\in \mathbb {R}$, let a real sequence be recursively defined by

${}x_{n+1}={\frac {x_{n}+a}{2}}\,.$

Show the following statements.

(a) For ${}x_{0}>a$, we have ${}x_{n}>a$ for all ${}n\in \mathbb {N}$, and the sequence is strictly decreasing.

(b) For ${}x_{0}=a$, the sequence is constant.

(c) For ${}x_{0}<a$, we have ${}x_{n}<a$ for all ${}n\in \mathbb {N}$. and the sequence is strictly increasing.

(d) The sequence converges.

(e) The limit is ${}a$.

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