Melting Ice: Does Water Surface Rise?

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When ice melts in a glass of water, the water level remains unchanged because the ice displaces an amount of water equal to its mass while floating. As the ice melts, it converts to water with the same mass, maintaining the water level. This principle is illustrated by the example of a boat: when a dense object is removed from the boat and submerged, the water level drops because it now displaces only its volume instead of its mass. The argument is supported by the physics of buoyancy and displacement. Therefore, the water surface does not rise when the ice melts.
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When you take bottle of glass and fill it with some ice what happens to watersurface when ice has melted away? Does it rise surface? Or will it remain the same? Teacher spoke and demonstrated that and it didn't rise the surface. I believe in my eyes but is there any physical way to prove that surface doesn't rise up?

This is not homework it's only asked because I'm arguing about it with my friend and he doesn't believe me..
 
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A floating object displaces its own mass in water. When it melts it is the same mass of water as it was when it was ice. Now it displaces its own volume as liquid water.

It might help if you think of another case. If a boat in a swimming pool has a relatively dense object in it the water level will drop if you take the object out of the boat and drop it into the water. The object was floating before and was displacing its mass of water while in the boat. When it is dropped into the water it displaces only the volume of the object.
 

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