Convergent Sequence from Introduction to Analysis Consider the sequence of real numbers
  $$\frac 12, \cfrac 1{2+\cfrac 1 2}, \cfrac 1{2+\cfrac 1{2+\cfrac 12}}, \ldots.$$ 
  
  Show that this sequence is convergent and find its limit by first showing that the two sequences of alternate terms are monotonic and finding their limits.


(From Introduction to Analysis by Maxwell Rosenlicht)

I am not sure if I have burned out or what, but I cannot do this. Any help would be much appreciated.
So do we have

$$x_{n+1} = {1\over 2 + x_n}?$$

If so, I can edit this answer to produce a solution.
It appears so.

So, in the steady state, we have $x_n = x_{n+1}$. We solve this equation

$$t= {1\over (2 + t)}.$$

Multiplying, 
$$ t^2 + 2t = 1.$$
so
$$ t^2 + 2t -1 = 0.$$
Availing ourselves of the quadratic formula we get
$$ t = {-2\pm \sqrt{4 + 4}\over 2} = -1\pm \sqrt{2}.$$
In this case, you will be attracted to the positive root $\sqrt{2} - 1$.
So do we have

$$x_{n+1} = {1\over 2 + x_n}?$$

If so, I can edit this answer to produce a solution.
It appears so.

So, in the steady state, we have $x_n = x_{n+1}$. We solve this equation

$$t= {1\over (2 + t)}.$$

Multiplying, 
$$ t^2 + 2t = 1.$$
so
$$ t^2 + 2t -1 = 0.$$
Availing ourselves of the quadratic formula we get
$$ t = {-2\pm \sqrt{4 + 4}\over 2} = -1\pm \sqrt{2}.$$
In this case, you will be attracted to the positive root $\sqrt{2} - 1$.