The point is that you consider " infinite " same as a " potential " number rather than an " actual " number
This was the ancient Greeks' idea of infinite
In the modern math infinite is considered as an ordinary number which can be manipulated same as any other number
You must not consider the series : 1/2 + 1/4 + ...
same as some thing unfinished
You must consider it same as any other normal math expression which supplies one and only one result
It is somewhat similar to the irrational number issue
The diagonal of a square having side 1 is : square(2)
Try to write this number
You can not , the number of figures after the decimal point being infinite
Does it mean that the diagonal of a square is infinite ?
Dont students use irrational numbers along with integer numbers ?
This is the logic explanation
As far as the Physical explanation is concerned, well actually it would not deserve even an explanation being evident that movement does exist regardless of the phylosophers' nonsenses
However
You claim that you have reached your goal when the ditance is : 0
But you must also measure such distance so you have to do with the measuring errors due to measuring istruments
Halving and halving your distance , sooner or later your super sophisticated istruments will tell you : hey you have reached your goal