Dichotomy Paradox

Paradox
Motion is impossible.

Suppose $B$ is a body which is to move from $P$ to $Q$.

Before $B$ can reach $Q$, it must first move halfway to $Q$.

But before it can reach the midpoint between $P$ and $Q$, it must first move halfway to that midpoint.

And before it can reach the point half way to the midpoint, it has to move halfway to that point too.

So on, indefinitely.

It is therefore impossible for motion to happen, because you cannot even get started. Every time you try to reach a point, you have to reach the point half way before it first.

Resolution

 * $\ds \sum_{n \mathop \ge 1} 2^{-n} = 1$