  We study dynamic cooling, where an externally driven two-level system is
cooled via reservoir, a quantum system with initial canonical equilibrium
state. We obtain explicitly the minimal possible temperature $T_{\rm min}>0$
reachable for the two-level system. The minimization goes over all unitary
dynamic processes operating on the system and reservoir, and over the reservoir
energy spectrum. The minimal work needed to reach $T_{\rm min}$ grows as
$1/T_{\rm min}$. This work cost can be significantly reduced, though, if one is
satisfied by temperatures slightly above $T_{\rm min}$. Our results on $T_{\rm
min}>0$ prove unattainability of the absolute zero temperature without
ambiguities that surround its derivation from the entropic version of the third
law. The unattainability can be recovered, albeit via a different mechanism,
for cooling by a reservoir with an initially microcanonic state. We also study
cooling via a reservoir consisting of $N\gg 1$ identical spins. Here we show
that $T_{\rm min}\propto\frac{1}{N}$ and find the maximal cooling compatible
with the minimal work determined by the free energy.
