Putnam 2016 B1

Let $x_0,x_1,x_2,\dots$ be the sequence such that $x_0=1$ and for $n \geq 0$, $$x_{n+1} = \ln(e^{x_n} - x_n)$$ (as usual, the function $\ln$ is the natural logarithm). Show that the infinite series $$x_0 + x_1 + x_2 + \cdots$$ converges and find its sum.