Does electrons keeps on removing from conductor under this setup?

Question

Let say we bring a conductor near a high static positively charged surface and connect earth to the side of conductor facing the positively charged surface. Then due to positively charged surface, let say conductor will generate electrostatic induction and 4 electron shift towards the positively charged surface(4 electron number is just taken to make you understand question in simple way). Connecting the wire from earth to conductor's face which is facing toward positively charged surface will cause electron to flow from conductor to earth. And once let say 4 electron is gone from conductor then by electrostatic induction other 4 electron from the behind inside the conductor comes on surface and again these 4 electron goes to earth. So by this process can conductor run out of electron completely?