How Do Limits Resolve Zeno’s Achilles Paradox?

Zeno’s Achilles paradox imagines a fast runner trying to catch a slower tortoise that has a head start. The runner must first reach the tortoise’s starting point; by then, the tortoise has moved on. The runner must then reach that new point, while the tortoise moves again, and so on without end.

The modern mathematical answer is that infinitely many stages can fit into a finite amount of time. The stages get shorter, and their durations form a convergent infinite series. Here is how the calculation works, and what it does—and does not—resolve.

The Achilles setup

Suppose Achilles runs at constant speed $v_A$, the tortoise runs at constant speed $v_T$, and $v_A>v_T$. Let the tortoise start a distance $d$ ahead. When Achilles covers that initial distance, the tortoise has moved forward. When Achilles covers that extra distance, the tortoise has moved again.

At each stage, the gap shrinks by the same ratio. To see why, the first stage lasts $d/v_A$ seconds. During that time, the tortoise travels $v_Td/v_A$, which is a fraction $v_T/v_A$ of the original gap. Call this ratio $r=v_T/v_A$. Since Achilles is faster, $0<r<1$.

The next stage therefore lasts $r$ times as long as the first. The stage durations are

$$\frac{d}{v_A},\quad \frac{d}{v_A}r,\quad \frac{d}{v_A}r^2,\quad \frac{d}{v_A}r^3,\quad\ldots$$

Add the infinite sequence of times

The total time represented by all these stages is a geometric series:

$$\begin{aligned} T&=\frac{d}{v_A}(1+r+r^2+r^3+\cdots)\\ &=\frac{d}{v_A}\frac{1}{1-r}\\ &=\frac{d}{v_A-v_T}. \end{aligned}$$

The formula $1+r+r^2+\cdots=1/(1-r)$ applies because $|r|<1$. More precisely, the sum is the limit of its finite partial sums. After $n$ stages, the partial sum is $1+r+\cdots+r^{n-1}=(1-r^n)/(1-r)$. As $n$ grows, $r^n$ approaches zero, so the partial sums approach $1/(1-r)$.

That finite limit is the time it takes the gap to close. It also agrees with the direct relative-speed calculation: the gap closes at speed $v_A-v_T$, so closing distance $d$ takes $d/(v_A-v_T)$ seconds.

A worked numerical example

Suppose the tortoise starts $10$ meters ahead. Achilles runs at $10$ meters per second, and the tortoise at $1$ meter per second. Achilles takes $1$ second to reach the tortoise’s starting point. During that second, the tortoise moves $1$ meter. Achilles takes $0.1$ second to cover that meter, during which the tortoise moves $0.1$ meter. The following stage takes $0.01$ second, then $0.001$ second, and so forth.

The total time for these stages is

$$1+0.1+0.01+0.001+\cdots=\frac{1}{1-0.1}=\frac{10}{9}\text{ seconds}.$$

So the infinitely many listed stages take a finite total time: about $1.111$ seconds. In that time Achilles travels $10\times 10/9=100/9$ meters. The tortoise travels $1\times 10/9=10/9$ meters, and its $10$-meter head start brings it to the same position. The two calculations agree.

Why an infinite number of stages is not a contradiction

Every finite stage leaves a small positive gap. But the gap after repeatedly applying the same reduction factor is proportional to $r^n$, and $r^n$ tends to zero. The stages do not need to have a “last” member. Instead, their times approach a particular finite limit, and the motion model says the runners meet at that time.

The same idea answers Zeno’s dichotomy version. To cross a distance $L$ at speed $v$, you can divide the route into halves, then quarters, then eighths, and continue. The travel times add to

$$\frac{L}{2v}+\frac{L}{4v}+\frac{L}{8v}+\cdots=\frac{L}{v}.$$

Dividing a finite journey into infinitely many smaller parts does not make its total duration infinite when those parts shrink quickly enough.

Historically, this is a modern mathematical explanation rather than a claim that Zeno himself used limits. Aristotle discussed the paradoxes in Physics, distinguishing an endless process of division from an actually completed infinity. Calculus and the rigorous theory of limits later supplied precise tools for summing the shrinking intervals.

Common mistakes

• Assuming every infinite sum diverges. Some do, such as $1+1+1+\cdots$. Others converge; a geometric series with ratio between $-1$ and $1$ is a standard example.
• Looking for a final stage. There is no last stage in the sequence. Convergence concerns the limit of all finite partial sums, not a final term.
• Treating the series calculation as a complete philosophical answer. It shows that Zeno’s sequence of required times is finite under the usual continuous-motion model. Broader questions about the nature of space, time, and motion require additional philosophical assumptions.
• Confusing the stages with separate delays. The stages are parts of one continuous chase. Their durations add to the elapsed time; they are not pauses that prevent Achilles from moving.