Discovering neutrino decay would be strong evidence of physics beyond the Standard Model. Presently, there are only lax lower limits on the lifetime $tau$ of neutrinos, of $tau/m sim 10^{-3}$ s eV$^{-1}$ or worse, where $m$ is the unknown neutrino mass. High-energy cosmic neutrinos, with TeV-PeV energies, offer superior sensitivity to decay due to their cosmological-scale baselines. To tap into it, we employ a promising method, recently proposed, that uses the Glashow resonance $bar{ u}_e + e to W$, triggered by $bar{ u}_e$ of 6.3 PeV, to test decay with only a handful of detected events. If most of the $ u_1$ and $ u_2$ decay into $ u_3$ en route to Earth, no Glashow resonance would occur in neutrino telescopes, because the remaining $ u_3$ have only a tiny electron-flavor content. We turn this around and use the recent first detection of a Glashow resonance candidate in IceCube to place new lower limits on the lifetimes of $ u_1$ and $ u_2$. For $ u_2$, our limit is the current best. For $ u_1$, our limit is close to the current best and, with the imminent detection of a second Glashow resonance, will vastly surpass it.
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