The series you mention is "Life of Mammals" by BBC, with David Attenborough. I saw the episode last week. (It has also been on discovery channel, but I can't remeber the name of that series)
The hunter actually ran after an atilope for 8 hours, finally the antilope was so hot and tired it could no longer move. He just walked right up to it and put the spear right into its heart.
It can be bought here:
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