Is there a finite unit of distance that we cannot divide past?

If distance could be divided into an infinite no of units or points, then it seems to me that motion would be impossible since a meter for instance, having an infinite no of points within it (and the points having an infinite point within it), would be un-breachable. It seems (intuitively) to me that for a measure of distance to be finite it would have to be made up of a finite unit of points however small those units were.

Otherwise, we are all teleporting at some point in the process of motion...

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The ancient Greek geometers had the same problem, and it all came down to this: they didn't fully appreciate that infinite sums can converge to finite values. – Chris White Jan 20 '13 at 2:47
@Chris White, good remark (+1). Bill Bryson says in a short column, "There is always a little more toothpaste in the tube, think about it!" and the underlying trick is also an infinite convergent series. – Eduardo Guerras Valera Jan 20 '13 at 3:35

You may find a better answer among philosophers than physicists for that question. But if you insist in getting a physical answer, it might be that there exists a length that is considered in the order of the shortest meaningful distance, called Plank Length, around $10^{-35}$ m. There are (physical) theoretical speculations about the possibility that, at that scales, space together with time could be sort of discrete.