How to probe $\lim\limits_{n\to\infty}\,\,{x_n}=a\in{R^m},\;$ if ${x_{n + 1}} = \frac{{a + {x_n}}}{2},\; \forall n \in \mathbb{N}$

Question

How to probe $\lim\limits_{n\to\infty}\,\,{x_n}=a\in{\mathbb{R}^m}\;\;$, $\quad$ if ${x_{n + 1}} = \frac{{a + {x_n}}}{2},\quad \,\forall n \in \mathbb{N}$

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I want to know how to delimit the expression $$\left\| {{x_n} - a} \right\|$$

Thansk for coments.

Answer 1

Hint $$x_{n+1}-a=\frac{1}{2} (x_n-a)$$

Second hint By Induction

$$x_{n+1}-a=\frac{1}{2^n} (x_1-a)$$