Boundedness and convergence of $x_{n+1} = x_n ^2-x_n +1$ Suppose that $x_0 = \alpha \in \mathbb{R}$ and $x_{n+1} = x_n ^2-x_n +1$.
I am asked to study the boundedness of $(x_n)$ and then asked if $(x_n)$ converges. How can I show that $(x_n)$ is bounded?
I have noted that $$x_{n+1}-x_n = (x_n-1)^2\geq 0$$ so $x_n$ is increasing. 
Suppose that $(x_n)$ converges, then $l = l^2-l+1$ so $l = 1$. This means that the sequence converges only when $\alpha \leq 1$. And so $(x_n)$ is bounded above by $1$ if $\alpha\leq 1$ (since an increasing sequence converges to its supremum) otherwise the sequence is not bounded above.
Is my reasoning correct?

Notice that $x_{n+1}=1-x_n(1-x_n)$. Thus $x_{n+1}\in[0,1]$ if and only if $x_n\in[0,1]$. Thus the sequence is bounded above by $1$ if and only if $\alpha\in[0,1]$.
