  We show that the Lieb-Liniger model for one-dimensional bosons with repulsive
$\delta$-function interaction can be rigorously derived via a scaling limit
from a dilute three-dimensional Bose gas with arbitrary repulsive interaction
potential of finite scattering length. For this purpose, we prove bounds on
both the eigenvalues and corresponding eigenfunctions of three-dimensional
bosons in strongly elongated traps and relate them to the corresponding
quantities in the Lieb-Liniger model. In particular, if both the scattering
length $a$ and the radius $r$ of the cylindrical trap go to zero, the
Lieb-Liniger model with coupling constant $g \sim a/r^2$ is derived. Our bounds
are uniform in $g$ in the whole parameter range $0\leq g\leq \infty$, and apply
to the Hamiltonian for three-dimensional bosons in a spectral window of size
$\sim r^{-2}$ above the ground state energy.
