I'm currently trying to demonstrate a method that I've seen that looks promising to extend tetration, but that's not directly the topic of this question. This question is part of the method, so part of the demonstration I need to do.
$f(x)=\log_{a}{(x+1)}$
I need to demonstrate that the infinite iteration of $f(x)$ for any $x≥0$ converges to 0 if and only if $a≥e$
$f(f(f(...x)))=0, ~\forall{x≥0}$ (If $a≥e$)
Through some tests, intuition and other factors I'm almost 100% sure this is correct, but to be mathematically rigorous I need to demonstrate it. I have a feeling that using fixed points could be a good way to demonstrate it, but I'm not 100% sure. If you have anything to say, feel free to answer.