Amended Proof of Pitaevskii-Stringari's Theorem

An amended proof of Pitaevskii-Stringari's theorem is given in mathematically coherent manner without resorting to the Bogoliubov approximation. This approach is based on the orthodox quantum field theory which rigorously maintains the canonical commutation relations. Moreover, we make a sound argument by taking the thermodynamic limit which the authors of the original papers did not refer to. We conclude that there is no Bose Einstein Condensation (BEC) as a phase transition based on the Spontaneous Symmetry Breaking (SSB) of the global $U(1)$ gauge symmetry in flat one dimensional geometry even absolute zero temperature.