  Asteroseismology of dwarf stars cooler than the Sun is very challenging due
to the low amplitudes and rapid timescales of oscillations. Here, we present
the asteroseismic detection of solar-like oscillations at 4-minute timescales
($\nu_{\mathrm{max}}\sim4300\mu$Hz) in the nearby K-dwarf $\sigma$ Draconis
using extreme precision Doppler velocity observations from the Keck Planet
Finder and 20-second cadence photometry from NASA's Transiting Exoplanet Survey
Satellite. The star is the coolest dwarf star to date with both velocity and
luminosity observations of solar-like oscillations, having amplitudes of
$5.9\pm0.8\,$cm$\,\text{s}^{-1}$ and $0.8\pm0.2$ ppm, respectively. These
measured values are in excellent agreement with established luminosity-velocity
amplitude relations for oscillations and provide further evidence that mode
amplitudes for stars with $T_{\mathrm{eff}}<\,5500\,$K diminish in scale
following a $(L/M)^{1.5}$ relation. By modeling the star's oscillation
frequencies from photometric data, we measure an asteroseismic age of
$4.5\pm0.9\,\rm{(ran)} \pm 1.0\,\rm{(sys)}$ Gyr. The observations demonstrate
the capability of next-generation spectrographs and precise space-based
photometry to extend observational asteroseismology to nearby cool dwarfs,
which are benchmarks for stellar astrophysics and prime targets for directly
imaging planets using future space-based telescopes.
