  We consider the well-known Lieb-Liniger (LL) model for $N$ bosons interacting
pairwise on the line via the $\delta$-potential in the mean-field scaling
regime. Assuming suitable asymptotic factorization of the initial wave
functions and convergence of the microscopic energy per particle, we show that
the time-dependent reduced density matrices of the system converge in trace
norm to the pure states given by the solution to the one-dimensional cubic
nonlinear Schr\"odinger equation (NLS) with an explict rate of convergence. In
contrast to previous work arXiv:0906.3047 relying on quantum field theory and
without an explicit rate, our proof is inspired by the counting method of Pickl
arXiv:0907.4464 and Knowles and Pickl arXiv:0907.4313. To overcome difficulties
stemming from the singularity of the $\delta$-potential, we introduce a new
short-range approximation argument that exploits the H\"older continuity of the
$N$-body wave function in a single particle variable. By further exploiting the
$L^2$-subcritical well-posedness theory for the 1D cubic NLS, we can prove
mean-field convergence assuming only that the limiting solution to the NLS has
finite mass.
