Distance or time?

Two brothers A and B have the exact same walking speed. They also have the same running speed, which of course is faster than the walking speed.

One day, they start a journey to the same destination at the same time. Brother A walks half the distance and runs the rest, whereas Brother B walks for half of the time and runs for the other half of the time.

Who will reach the destination first?

Challenge: Prove your answer in a formal manner.