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The paradox I am referring to is that which can be resolved by considering the fact that ##\sum_{k=0}^\infty1/2^k=1##. However, before one can travel half of the distance to be travelled, he has to travel half of that half, and half of that half ... Moreover, to say that one can travel by halves implies that he can travel by thirds and fourths ... Therefore, if one can even begin to travel, he has to move ##1/n## of the distance, where ##n\to\infty##, but that is just ##0##, so one cannot start his travelling.

Is there any mistake in the reasoning?

Notice that shifting the problem to traveling the ##1/n,\,n\to\infty## of the distance gets you, once more, to the original problem.

The fact that you can travel ##0## distance doesn't allow you to say that you can go ## 0+ϵ## forward. I am not affirming that you can travel half, a fourth, or a third of a distance, rather what I am saying is that if you are saying that you can travel ##1/n##, then you surely can go ##1/(n+1)##, but can you prove that? You then continue downwards until you reach ##0## which I won't object you can travel, but how can you go upwards? (since I would reason in the same way again)

Is there any mistake in the reasoning?

Notice that shifting the problem to traveling the ##1/n,\,n\to\infty## of the distance gets you, once more, to the original problem.

The fact that you can travel ##0## distance doesn't allow you to say that you can go ## 0+ϵ## forward. I am not affirming that you can travel half, a fourth, or a third of a distance, rather what I am saying is that if you are saying that you can travel ##1/n##, then you surely can go ##1/(n+1)##, but can you prove that? You then continue downwards until you reach ##0## which I won't object you can travel, but how can you go upwards? (since I would reason in the same way again)

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