Free complement (FC) method provides a general and systematic method of solving the Schrödinger equation. In this method, the Hamiltonian of the system modified for the singularity of the potential is used to generate the FC functions that span the exact wave function of the system. Thus, by applying the variation principle to the sum of the complement functions, which we call FC wave function, we can calculate the essentially exact wave function and energy for the ground and excited states of the system. We here show that the Schrödinger equation can be solved to an arbitrary accuracy with the FC method by examining the upper and lower bounds of the energy, local energy, H-square error, cusp condition, and so on, for the helium atom.