  We report measurements of radon progeny in liquid argon within the MicroBooNE
time projection chamber (LArTPC). The presence of specific radon daughters in
MicroBooNE's 85 metric tons of active liquid argon bulk is probed with newly
developed charge-based low-energy reconstruction tools and analysis techniques
to detect correlated $^{214}$Bi-$^{214}$Po radioactive decays. Special datasets
taken during periods of active radon doping enable new demonstrations of the
calorimetric capabilities of single-phase neutrino LArTPCs for $\beta$ and
$\alpha$ particles with electron-equivalent energies ranging from 0.1 to 3.0
MeV. By applying $^{214}$Bi-$^{214}$Po detection algorithms to data recorded
over a 46-day period, no statistically significant presence of radioactive
$^{214}$Bi is detected, and a limit on the activity is placed at $<0.35$ mBq/kg
at the 95% confidence level. This bulk $^{214}$Bi radiopurity limit -- the
first ever reported for a liquid argon detector incorporating liquid-phase
purification -- is then further discussed in relation to the targeted upper
limit of 1 mBq/kg on bulk $^{222}$Rn activity for the DUNE neutrino detector.
