A sequence of real numbers $\displaystyle {U_n}$ satisfies the recurrence relation

$\displaystyle U_{n+1}=\frac{1}{2}U_n+1$ for all positive integer $\displaystyle n$.

a) Given that $\displaystyle {U_n}$ converges to $\displaystyle l$, find the value of $\displaystyle l$

b) If $\displaystyle U_n < l$, show that $\displaystyle U_n<U_{n+1}<l$

I was able to find that $\displaystyle l=2$ for part a.

For (b), what I have done is:

$\displaystyle 2U_{n+1}-2=U_n$

$\displaystyle 2U_{n+1}-2<2$

$\displaystyle U_{n+1}<0$

How do I continue? Or am I going in the wrong direction?