  We show that the weak limit of a quasiminimizing sequence is a quasiminimal
set. This generalizes the notion of weak limit of a minimizing sequences
introduced by De Lellis, De Philippis, De Rosa, Ghiraldin and Maggi. This
result is also analogous to the limiting theorem of David in local Hausdorff
convergence. The proof is based on the construction of suitable deformations
and is not limited to the ambient space $\mathbb{R}^n \setminus \Gamma$, where
$\Gamma$ is the boundary. We deduce a direct method to solve various Plateau
problems, even minimizing the intersection of competitors with the boundary.
Furthermore, we propose a structure to build Federer--Fleming projections as
well as a new estimate on the choice of the projection centers.
