Defending the Indefensible by Attacking the Obvious
Zeno of Elea was Parmenides' student, and his paradoxes are usually misunderstood. He was not trying to convince anyone that motion is impossible. He was arguing dialectically: his opponents said Parmenides' position was absurd, so Zeno showed that their position — that there are many things, and that they move — leads to absurdities just as bad. If both views are paradoxical, ridiculing Parmenides is no longer an argument.
The Dichotomy: to reach the end of a path you must first reach the halfway point, then half the remainder, and so on without end. Completing infinitely many tasks in finite time seems impossible, so motion cannot begin.
Achilles and the Tortoise: give the tortoise a head start. By the time Achilles reaches where it was, it has moved a little further. Repeat forever. Achilles never passes it.
The Arrow: at any instant, a flying arrow occupies a space exactly its own size, and so is at rest. Time is composed of instants. So the arrow is at rest at every instant of its flight.
The modern resolution of the first two is that an infinite series can have a finite sum:
Infinitely many terms, finite total. The intervals shrink fast enough that the whole journey takes finite time. This is the geometric series , and it converges, a fact Zeno had no way to state.
Whether that fully answers him is still argued. It shows the time is finite; whether it explains how infinitely many tasks are completed is a separate question, and the Arrow — which is about what motion consists in at an instant, not about summation — resists the treatment entirely.
Melissus defended the same position by different means, arguing that what-is must be spatially infinite (since a limit would be a boundary with what-is-not) and therefore incorporeal, since a body would have parts. He also gave the argument that if things really changed, then what-is would become what-is-not, so the very possibility of change refutes itself.
Common pitfall: thinking Zeno was refuted the moment convergent series were understood. The mathematics settles that the sum of the times is finite. It does not obviously explain how a supertask, infinitely many distinct completions, is performed, and the Arrow paradox is untouched by it.