TITLE: Why does $ a_n = \frac {a_{n-1} + \frac {2}{a_{n-1}}}{2}$ converge to an irrational number?
QUESTION [16 upvotes]: There is a problem in my textbook that goes like this
$$ a_n = \frac {a_{n-1} + \frac {2}{a_{n-1}}}{2}$$ 
and 
$$a_0 =1$$ 
for all $n\ge1$.
It is monotonically decreasing sequence of rational numbers and bounded below. However, it cannot converge to a rational number.
Then the task is to find the limit. The problem itself is easy but I don't understand how the author judged the limit to be irrational even before solving the question? Is there any property or did they just know the answer beforehand?

REPLY [5 votes]: Rewrite $$a_n = \frac {a_{n-1} + \frac {2}{a_{n-1}}}{2}$$ as $$a_n=a_{n-1}-\frac 12a_{n-1}+\frac 1{a_{n-1}}=a_{n-1}-\frac{a^2_{n-1}-2 }{2a_{n-1}}$$ and recognize the Newton formula for the solution of $x^2-2=0$.