Show that for some choice of $x_1\in\mathbb{R}$, the sequence given by: $$ x_{n+1} = n-3^{x_n} $$ satisfies: $$ \lim_{n\to +\infty} x_n= +\infty. $$
I was able to prove the statement by showing that the odd-looking $$ x_n = \log_3\big(n-\log_3\big(n+1-\log_3\big(n+2-\ldots$$ is well defined. Are you aware of other effective techniques?
I am especially interested in topological ones.