Why does time slow down for someone standing at the bottom of a mountain compared to time for someone at the top of a mountain?

The concept of time is influenced by the force of gravity. On a planet, gravity acts on every object within its gravitational field. The closer the object is to the surface of the planet, the greater influence gravity has on that object. Therefore someone standing at the bottom of a mountain is influenced by gravity more than a man at the top of the mountain which means that time slows down more for the person at the bottom of the mountain as they are closer to the surface of the planet.