We have $f,g: \mathbb{R} \to \mathbb{R}$ two contractions with the same fixed point and $(x_{n})_{n\geq1}$ a sequence with real numbers with the propriety that $x_{n+1}\in \{f(x_{n}),g(x_{n})\}$, for any $n\geq1$. Show that $(x_{n})_{n\geq1}$ is convergent.
I was thinking to solve in 3 cases. First case when $x_{n+1} = f(x_{n})$ only, the second case when $x_{n+1} = g(x_{n})$ only and the last case when $x_{n+1} = f(x_{n})$ or $x_{n+1} = g(x_{n})$, when the sequence has alternating recursion with unknown periodicity. I was thinking breaking the sequence in 2 subsequences $x_{k_{m}}$ and $x_{q_{p}}$, where the sequence $k_{m}$ is the sequence of the positions (when the function f is in the recursion) and $q_{p}$ is the sequence of the positions (when the function $g$ is in the recursion) and for a sequence with "$n$ terms" we will have "$m$ terms" for the first subsequence + "$p$ terms" for the second subsequence. For the first two cases the problem is easily solved using Banach Theorem for fixed point. What should I use for the third case? The definition of contraction?