Show that the sequence $(x_n)_{n\in\mathbb{N}}$ defined by $x_1=3/4$ and $x_{n+1}=\frac{x_n^2+x_n}{2}$ for $n\geq 1$ is convergent. Evaluate the limit of the sequence.
I was trying to prove the sequence is a Cauchy sequence and therefore convergent, but I couldn't. I think proving it's Cauchy might be over complicating things. Any input is welcome!
Since $0<x_1<1$ and $0<x_n<1 \implies 0<x_{n+1}<x_n$ by induction the sequence is both decreasing and bounded, and by the monotone convergence theorem it has a limit. Call it L. Taking the limit in the definition of the sequence gives $$ L = \frac{L^2+L}2 \implies L=0 \text{ or } 1. $$ Since the sequence is decreasing, $L=0$.