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Conservation of energy and the electron-degenerate pressure. For the neutron to decay you must have $$n \to p + e^- + \bar{\nu}$$ or $$n + \nu \to p + e^- \quad.$$ In either case that electron is going to stay around, but in addition to the neutrons being in a degenerate gas, the few remaining electrons are also degenerate, which means that adding a ...
There is Beta decay in neutron stars. This is the simple answer. Since a neutron star is electrical neutral, there is the same amount of $\beta^+$ as $\beta^-$ decay, this is called the chemical equilibrium. This means, every time when a neutron decays, a proton captures (in average) an electron and the star stays stable.