  Thermodynamic uncertainty relations (TURs) bound the dissipation in
non-equilibrium systems from below by fluctuations of an observed current.
Contrasting the elaborate techniques employed in existing proofs, we here prove
TURs directly from the Langevin equation. This establishes the TUR as an
inherent property of overdamped stochastic equations of motion. In addition, we
extend the transient TUR to currents and densities with explicit
time-dependence. By including current-density correlations we, moreover, derive
a new sharpened TUR for transient dynamics. Our arguably simplest and most
direct proof, together with the new generalizations, allows us to
systematically determine conditions under which the different TURs saturate and
thus allows for a more accurate thermodynamic inference. Finally we outline the
direct proof also for Markov jump dynamics.
