Consider the sequence of functions $(f_n)_{n\geq 2}$ defined on $(0,1]$ by $$f_n(t)=\ln(t)-t+n.$$ By intermediate value theorem, we deduce that there exists a sequence $(t_n)_n$ with $0<t_n<1$ such that $f_n (t_n)=0$ for all $n\geq2$. Thus we have $$0<t_n=\ln(t_n)+n<1$$ and we can then prove that $$e^{-n}<t_n<e^{1-n}$$ which implies that $(t_n)_n$ converges to $0$. Using Mathematica I can see that $(t_n)_n$ is decreasing. I don't know how we can prove that. I would be also glad to find the speed of convergence to $0$, which by the above inequality would be something between $e^{-n}$ and $e^{1-n}$.
Here's the graph of those functions for $n=2,3,4$
We have $ n = t_n - \log t_n,$ so $e^n = e^{t_n}/t_n,$ and $e^{t_n+1}/t_{n+1}=e^{n+1}=e \cdot e^{t_n}/t_n > e^{t_n}/t_n.$
Taking derivatives, we find that $g(t) = e^t/t$ decreases on $0<t<1,$ which proves the theorem.