zeno's paradox revisited

I like to solve paradoxes while trying to sleep. Sometime when I don't have a paradox to work on, I go back over old paradoxes to make sure I know how to solve them.

Recently I revisited Zeno's paradox. I usually think of the paradox as trying to catch up with some moving thing, which first requires getting to where it is now, which takes some time, and then getting to where it went during that time, which takes more time, and then getting to where it went during that time, and so on.

I'm usually satisfied with the argument that although there are an infinite number of imaginary way-points to get to, they will be gotten to more and more rapidly, such that the total time it takes to catch up with the moving thing is limited.

However, I got tripped up on the following idea: what if we actually step to the first way-point with our right foot, and then step to the next way-point with our left foot, and the next with our right, and so on.

In this case, will we ever catch up with the moving thing, and if so, what foot will we land on when we do?

One could argue that at some point it will become physically impossible to take a step of the required length at the required speed, but in principle, it seems physically possible to carry out this experiment. So in principle, what would happen?

My current leaning is that if space-time were not quantized, this would actually be an issue.

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