Can we use the same water to submerge the arms and feet the three times in wudu?   

In the Name of Allah, Most Compassionate, Most Merciful,

When submerging one’s arms (for example) in water:

1. If the volume of water that comes in contact with one’s arm is more or equal to the volume of the rest of the water, all of the water will be considered used (musta’mal), and will no longer have the ability to purify.
    
2. If the water that comes into contact with one’s arm is less than the rest of the water, then all of the water will remain unconditioned (mutlaq), and can be used again to purify.

Imam al-Tahtawi says: “So the upshot is, that it is permissible to make the ritual bath and ablution from small pools, as long as one is not reasonably sure that the used water is more or equal (to the rest of the water), and that one is not reasonably sure that filth (najasa) fell into it.” [Hashiyat al-Tahtawi ‘ala Maraqi al-Falah]

It would be more precautionary to wash the limbs by scooping the water with one’s hand than immersing the limbs in the water.

And Allah knows best

Fadi Qutub