This 'Zenon paradox' was created to tackle the sophists, to give them a problem to go against (sophists job was to 'create truth' within any given situation, they had to be right! I guess the new sophists are politicians)
The problem with this is that you consider distance like an infinite thing while it isn't infinite. A string has a length, and after a while only moving for the smallest thing will not be the half of the remaining distance...