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I need to demonstrate that U(x) = phi(x) + x*Phi(x) > 0 for all x.  This is clearly true for x >= 0.  In addition U'(x) > 0 for all x, and thus it is a 

strictly increasing function.

It would be sufficient to show, for instance that lim (x-> - infinity) U(x) = 0

Any suggestion ?

Thanks in advance,

Esteban