This integral may be evaluated by defining the wave function over the complex E plane and closing the E contour using a semicircle on which the wavefunctions vanish.  The integral over the closed contour may then be evaluated, using the Cauchy integral theorem, as a sum of the residues at the various poles.  We will now argue that the residues of  approach those of  at time  and so the corresponding wavepackets are equal at temporal infinity.
In fact, for very positive times t the  factor in a Schrödinger picture state forces one to close the contour on the lower half-plane.  The pole in the  from the Lippmann–Schwinger equation reflects the time-uncertainty of the interaction, while that in the wavepackets weight function reflects the duration of the interaction.  Both of these varieties of poles occur at finite imaginary energies and so are suppressed at very large times.  The pole in the energy difference in the denominator is on the upper half-plane in the case of , and so does not lie inside the integral contour and does not contribute to the  integral.  The remainder is equal to the  wavepacket.  Thus, at very late times , identifying  as the asymptotic noninteracting out state.
Similarly one may integrate the wavepacket corresponding to  at very negative times.  In this case the contour needs to be closed over the upper half-plane, which therefore misses the energy pole of , which is in the lower half-plane.  One then finds that the  and  wavepackets are equal in the asymptotic past, identifying  as the asymptotic noninteracting in state.