  This is the main paper of a series establishing the linear stability of
Schwarzschild-Anti-de Sitter (AdS) black holes to gravitational perturbations.
Specifically, we prove that solutions to the linearisation of the Einstein
equations $\textrm{Ric}(g) = \Lambda g$ with $\Lambda<0$ around a
Schwarzschild-AdS metric arising from regular initial data and with standard
Dirichlet-type boundary conditions imposed at the conformal boundary (inherited
from fixing the conformal class of the non-linear metric) remain globally
uniformly bounded on the black hole exterior and in fact decay inverse
logarithmically in time to a linearised Kerr-AdS metric. The proof exploits a
hierarchical structure of the equations of linearised gravity in double null
gauge and crucially relies on boundedness and logarithmic decay results for the
Teukolsky system, which are independent results proven in Part II of the
series. Contrary to the asymptotically flat case, addition of a residual pure
gauge solution to the original solution is not required to prove decay of all
linearised null curvature and Ricci coefficients. One may however normalise the
solution at the conformal boundary to be in standard AdS-form by adding such a
pure gauge solution, which is constructed dynamically from the trace of the
original solution at the conformal boundary and quantitatively controlled by
initial data.
