  I provide evidence that the 2D $RP^{N-1}$ model for $N \ge 3$ is equivalent
to the $O(N)$-invariant non-linear $\sigma$-model in the continuum limit. To
this end, I mainly study particular versions of the models, to be called
constraint models. I prove that the constraint $RP^{N-1}$ and $O(N)$ models are
equivalent for sufficiently weak coupling. Numerical results for their
step-scaling function of the running coupling $\bar{g}^2= m(L) L$ are
presented. The data confirm that the constraint $O(N)$ model is in the samei
universality class as the $O(N)$ model with standard action. I show that the
differences in the finite size scaling curves of $RP^{N-1}$i and $O(N)$ models
observed by Caracciolo et al. can be explained as a boundary effect. It is
concluded, in contrast to Caracciolo et al., that $RP^{N-1}$ and $O(N)$ models
share a unique universality class.
