  We clarify basic properties of an effective action (i.e., self-consistent
perturbation expansion) for interacting Bose-Einstein condensates, where field
$\psi$ itself acquires a finite thermodynamic average $\langle \psi\rangle$
besides two-point Green's function $\hat G$ to form an off-diagonal long-range
order. It is shown that the action can be expressed concisely order by order in
terms of the interaction vertex and a special combination of
$\langle\psi\rangle$ and $\hat G$ so as to satisfy both Noether's theorem and
Goldstone's theorem (I) corresponding to the first proof. The self-energy is
predicted to have a one-particle-reducible structure due to $\langle
\psi\rangle\neq 0$ to transform the Bogoliubov mode into a bubbling mode with a
substantial decay rate.
