Convergence of $x_n=\sqrt{2(x_{n-1}-\log(1+x_{n-1}))}$

Question

$x_n=\sqrt{2(x_{n-1}-\log(1+x_{n-1}))}$ with $x_0=a\ge0$

I calculated some values for $x_n$ with $a=0,1,2$ and recognized its monotonically decreasing (It converges to $0$). Now the standard approach is to prove the monotonicity and that its bounded from below.

Do you think there is a more elegant way or is itsomehow possible to reduce it to a different recurrence?

Answer 1

What wrong with using the monotone convergence theorem? I think that is a very elegant way of proving convergence. Note that if $x \ge 0$ then $0 \le 2(x-\log(1+x)) \le x^2$ so that $0 \le x_n \le x_{n-1}$ for all $n$.