The Muller sequence is given by the recursive definition:
$U(n+1)=111-\frac{1130}{U(n)}+\frac{3000}{U(n)U(n-1)}$ with $U(0)=5.5$ and $U(1)=\frac{61}{11}$.
This sequence is interesting in floating-point arithmetic because it fails to converge numerically while it converges theoretically and has the limit 6 (more info).
But, how to prove theoretically that this sequence converges to its limit?
My advice is to read the paper claiming this, I couldn't find a source other than your link. I haven't dealt with such sequences before, but it does sound strange that there exists a limit. Namely, if a limit $L=\lim_{n\to\infty} U(n)$ exists, it is the same as $\lim_{n\to \infty} U(n+1)$ and $\lim_{n\to\infty} U(n-1)$. So the recursive definition gives a cubic equation in $L$ whose solutions are $5,6,100$, making me think there's no limit at all.