I will show that any big line bundle on a smooth projective variety admits a special Fujita approximation: the volume and the first Riemann-Roch coefficient are both approximated by those of $\mathbb{Q}$-ample line bundles on higher models. As already known by previous works, I will then explain how this implies a solution to the Yau-Tian-Donaldson conjecture, connecting the $K$-stability of a smooth polarized projective variety $(X,L)$ to the existence of constant scalar curvature Kähler metrics in $c_1(L)$.