Could you chase down a deer?
There are many animals that can run faster than humans.
There are not many that can run as far.
An incredible video of persistance hunting from the BBC's The Life of Mammals episode 10: Food for Thought.
"...we began adding meat to our diets around 2.6 million years ago, long before we developed advanced weapons like the bow and arrow, which was developed as recently as 50,000 years ago."
"We can run in conditions that no other animal can run in."
"Most animals would develop hyperthermia -- heat stroke in humans -- after about 10 to 15 kilometers" (6-10 miles)
So we began hunting by chasing our prey until it collapsed: Persistence Hunting.