Proving convergence of a sequence Let the following recursively defined sequence:
$a_{n+1}=\frac{1}{2} a_n +2,$
$a_1=\dfrac{1}{2}$.
Prove that $a_n$ converges to 4 by subtracting 4 from both sides.
When I do that, I get:
$2(\frac{1}{2} a_{n+1} -2)=(\frac{1}{2} a_n -2)$, so $y=2y$, 
which is true only for $0$. But I'm not sure how to formally use this in a definition of convergence?

Set $b_n=a_n-4$. Then $b_1=-7/2$ and 
$$
b_{n+1}=a_{n+1}-4=\frac{1}{2}a_n+2-4=\frac{1}{2}(a_n-4)=\frac{1}{2}b_n.
$$ 
Thus
$$
b_n=\frac{b_{n-1}}{2}=\frac{b_{n-2}}{2^2}=\cdots=\frac{b_{1}}{2^{n-1}}=-\frac{7}{2^n},
$$
and finally
$$
a_n=4-\frac{7}{2^n}.
$$
Hence
$$
\lim_{n\to\infty} a_n=4.
$$
