Let $x_0=1$ and $$x_{k+1} = (1-a_k)\left(\frac{3}{2}-\frac{1}{2}\frac{1}{x_k}\right)$$ where $a_n$ is a known sequence satisfying that $a_k\in(0,1)$ for all $k$ and $a_k\to 0$ as $k\to\infty$. How to prove that $x_k\to 1$ as $k\to\infty$?

The difficulty here is that

- It is not known how fast $a_k$ converges to zero, and I don't know how it affect the convergence of $x_k$;
- $x_k$ may change sign and is not monotone, so I don't know how to prove $x_k$ even converges;
- Furthermore, if we assume $x_k$ do converge to some limit $x^*$, then by taking the limit, $$x^*=(1-0)\left(\frac{3}{2}-\frac{1}{2}\frac{1}{x^*}\right)$$ I find there are two possible solution $x^*=1/2$ or $x^*=1$. How to exclude the case that $x^*=1/2$?