Let $\displaystyle f: [0,1] \rightarrow [0,1] $ be an isometry in a metric space $\displaystyle ([0,1], d_{Eucl})$. Suppose $\displaystyle f(0)=0, \ \ \ x\in [0,1]$

Could you explain to me why the sequence $\displaystyle (f^n(x))_{n\geq1}$ is monotone and why does it converge with respect to the usual topology of $\displaystyle [0,1]$ and also with respect to $\displaystyle d$ and, in particular, it satisfies the condition of Cauchy?

I know that since isometry is a continuous and bijective function, it must be monotone. We put $\displaystyle f(0)=0$, so it must be increasing. But I can't figure out why $\displaystyle f^n(x) \le f^{n+1}(x)$

Could you help me with that? I would really appreciate a thorough explanation, because I can't find anything about isometry composition anywhere.

Thank you.