We present the detection of three super-Earths transiting the cool star LP415-17, monitored by K2 mission in its 13th campaign. High-resolution spectra obtained with High Accuracy Radial velocity Planet Searcher-North/Telescopio Nazionale Galileo (HARPS-N/TNG) showed that the star is a mid-late K dwarf. Using spectral synthesis models, we infer its effective temperature, surface gravity, and metallicity, and subsequently determined from evolutionary models a stellar radius of 0.58 R⊙. The planets have radii of 1.8, 2.6, and 1.9 R⊕ and orbital periods of 6.34, 13.85, and 40.72 d. High-resolution images discard any significant contamination by an intervening star in the line of sight. The orbit of the furthest planet has radius of 0.18 au, close to the inner edge of the habitable zone. The system is suitable to improve our understanding of formation and dynamical evolution of super-Earth systems in the rocky-gaseous threshold, their atmospheres, internal structure, composition, and interactions with host stars.