how can I prove it is a convergent sequence and its limit is $\sqrt{a}$ Let we have the following sequence $$x_1=1$$
$$x_{n+1}=\frac{x_n^2+a}{2x_n}$$ such that $$a>0$$ how can I prove it is a convergent sequence and its limit is $\sqrt{a}$

Calculate $ x_{n+1} - \sqrt{a} $, 
$$ x_{n+1} - \sqrt{a} = \frac{(x_n - \sqrt{a})^2}{2x_n},$$
and 
$$ x_2 = \frac{a + 1}{2} \geqslant \sqrt{a}.$$
Thus by induction we can prove that $\forall n \geqslant 2,\,\, x_n \geqslant \sqrt{a}$.
Now that 
$$ x_{n+1} - \sqrt{a} = \frac{x_n - \sqrt{a}}{2x_n} \cdot (x_n - \sqrt{a}) = \frac12\left(1 - \frac{\sqrt{a}}{x_n}\right) \cdot (x_n - \sqrt{a})$$
and we know that 
$$ 0 < \frac{\sqrt{a}}{x_n} \leqslant 1.$$
So 
$$ 0 \leqslant \frac12\left(1 - \frac{\sqrt{a}}{x_n}\right) < \frac12$$
and the sequence $\{x_n - \sqrt{a}\}$ converges to $0$ because
$$ \forall n > 2, \,\, 0 \leqslant x_n - \sqrt{a} < \frac{1}{2^{n-2}} (x_2 - \sqrt{a}).$$
