Rivers don't run out of water because of its source of water. Water from the ground, rivers and ocean evaporates and forms clouds. These clouds move toward the mountains where it rains or snows, depending on the temperature. The rain water and melting snow flows down the mountain and ultimately forms into rivers. Since this process of the earth and water occurs continually, there is always a flow of water down the mountain and the river is constantly supplied with a fresh stream of water.
This water will never end as long as there is precipitation due to the earth's natural ability to follow the scientific patterns. It is these rivers that have been a source of water, and commerce for people of ancient days. Thus historically, cities were often built near flowing rivers and have been a life source for cities and the people who live within them.