JHK images with angular resolutions approaching the diffraction limit of the 3.6 m Canada-France-Hawaii Telescope are used to investigate the bright asymptotic giant branch (AGB) content of the M31 bulge. The AGB tip in a field 26 from the galaxy center occurs at K = 15.6, which is significantly fainter than measured in previous ground-based studies that sampled similar projected distances from the center of M31 but were affected by crowding. Within 26 of the center of M31, the number density of bright AGB stars scales with r-band surface brightness and the K brightness of the AGB tip does not vary measurably with radius. It is concluded that the infrared bright AGB stars (1) belong to the bulge and not the disk and (2) are well mixed throughout the inner bulge, suggesting that they formed at a time when the overall structural properties of the M31 bulge were imprinted. The bolometric luminosity functions (LFs) of the M31 bulge and Baade's window are in excellent agreement, while the brightest AGB stars in the M31 bulge, the Galactic bulge, and M32 have similar MK. Barring a fortuitous tuning of age and metallicity to produce AGB tips with similar brightnesses, it is suggested that the brightest stars in M32 and the bulges of M31 and the Milky Way belong to an old, metal-rich population; these stars are bright not because they have a young or intermediate age, but because they have a high metallicity.
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