I have the following exercise:
Given 2 arbitrary real numbers $x_1 < x_2$ and given the sequence $x_n = \frac{1}{2}(x_{n-1}+x_{n-2})$ prove that it is contractive and find its limit.
What I did is this:
$|x_n-x_{n-1}|= |\frac{1}{2}(x_{n-1}+x_{n-2}) -x_{n-1}| = |\frac{1}{2}(x_{n-2} -x_{n-1})|$, which would imply that $\frac{1}{2} = C$ right?
Knowing that it converges we can call the limit A and take $x_n = \frac{1}{2}(x_{n-2} +x_{n-1})$ is equivalent to $ A = \frac{1}{2}(A +A) = A $, but that could be any number so I don't know how to find the limit.
