I heard Zeno's "Achilles and the Tortoise" paradox is resolved using the limit concept of mathematics, i.e., the time cost in every sub-race is added up, which has a limit, not limitless, so Achilles can catch up the Tortoise finally.
That resolution is based on two assumptions:
The second assumption can be strictly proved in math. But how to prove the first assumption?
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Imagine you are set the following homework problem:
Problem. Achilles and the Tortoise are moving along the same straight line in the same direction. Achilles is moving at 10 m/s and the Tortoise 0.1 m/s. At $t=0$ the Tortoise is 100 m ahead of Achilles. Let $T$ be the catch-up time, i.e. the time at which Achilles and the Tortoise meet. What is the value of $T$?
Solution 1. At $t=T$, Achilles will have travelled 100 m more than the Tortoise. Distance = speed $\times$ time, so we arrive the equation $10 \cdot T = 0.1 \cdot T + 100$. We rearrange to get $T=\frac{100}{9.9}\,\text{s}$.
Solution 2. Let's partition $T$ into parts and add them up. Let $T_1$ be the time at which Achilles has reached the Tortoise's starting point (so $T_1=10$) and in general let $T_{n+1}$ be the time elapsed between Achilles moving from his position at $t = \sum_{i=1}^n T_i$ to the Tortoise's position at $t = \sum_{i=1}^n T_i$. Then $T_{n+1} = \frac{0.1\cdot T_n}{10}=10^{-2} \cdot T_n$ and a straightforward induction shows that $T_n = 10^{3-2n}$. We can use the formula for a geometric series to add up the infinite sum: \begin{equation*} T = \sum_{i=1}^\infty T_i = \sum_{i=1}^\infty 10^{3-2i} = 10^3\cdot\lim_{n\to\infty} \sum_{i=1}^n\left(10^{-2}\right)^i = 10^3 \cdot \left(\frac{1}{1-10^{-2}}-1\right) = \frac{100}{9.9} \,\text{s} \end{equation*}
Remarks. I chose particular numbers (100m, 10 m/s, 0.1 m/s) in order to make the calculations easier and more readily understandable, but the calculations straightforwardly generalise to prove both assumptions given in the OP's question. Solution 2 is of course Zeno's paradox but without any mention of the word 'paradox'. It is a legitimate way to partition $T$, in the same way that it is legitimate to partition unity and write $1=0.\dot{9} = \sum_{i=1}^\infty \frac{9}{10^i}$. Lastly, note that Zeno's paradox talks about Achilles always being behind the Tortoise. The word 'always' is a temporal adverb and in the context of the paradox time is $T_i$, which vanishes. There is no "outside" time.
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Both Achilles and the tortoise pass through a continuum of points. Over any interval, there are infinitely points that each must pass through. Each point is of zero length, and so it takes no time to pass across any single point.
So, yes, Achilles must pass through the half-way point then half-half-way point, etc. and infinitely many such defined points, and this is not a problem.
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I'll add an answer that will sound cheeky but, I insist, hits the point:
You're asking for a reason why "the catch-up time equals the sum of a series".
Well:
A) Time is measured in real numbers.
B) Real numbers are well-defined in mathematics: They are, in one disguise or another, "certain sums of series". That's why we define real numbers the way we do, most commonly by Cauchy sequences or Dedekind cuts, which both can be translated to "this real number is (among many other things) this sum of a series".
To philosophers and physicists, the exact mathematical definitions of the real numbers might seem technical; but we cherish and laud them as an important breakthrough, because they are consistent and make the paradox disappear. Note the two ways to compute the time that Achilles passes the Tortoise in dwolfeu's answer: They give a consistent result based on that consistent theory of real numbers. It's the very definition of $\mathbb R$ which makes the results of the two computations, one seemingly purely algebraic, the other seemingly relying on a limit from calculus, identical.
One can, of course, doubt A). One could say, nobody has ever measured a time interval of $12\pi$ seconds or $\sqrt {15}$ hours (or: maybe sometimes it should come out as $-4i$ days). Fair enough (although then I would add, has anyone ever truly and beyond all doubts measured a time interval of $\frac{15}{17}$ minutes either?). However, in all actual experiments, all "Achilleses" have actually overtaken all "Tortoises" at times which to the precision of measurement possible agreed with what the mathematical model of real numbers gave out. And, honestly, what more could one want?
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Short answer: When two procedures used to solve a problem lead to different results (a paradox), the solution to the paradox is to show why one of the procedures is wrong. It is NOT showing why the wrong procedure must, would, or should lead to the same result. A complete explanation, and the solution, are given below.
What Does Solving a Paradox Mean?
As mentioned at the beginning of my article (at: https://bit.ly/2IM76rF ), a paradox proposes the existence of two different results as a solutions for the same problem. These results are inconsistent with each other, depending on which procedure is used. Only one can be correct. As Brown and Moorcroft suggest, we are not looking for a mathematical demonstration that Achilles reaches the tortoise. Assuming they are both running in the same direction, we know he will. We can calculate the exact time, given the distance between the two and the two speeds, using a simple formula:
t = distance/(difference in velocity)
Instead, explaining or invalidating a paradox is to show a fault in the paradox formulation, or the proposed solving procedure, so that we can exclude this procedure and demonstrate that there is only one result for the original problem. The solution of a paradox is the answer to the question: “How does the paradox formulation misrepresent reality or logic?” That is, we need to show why the proposed method is conceptually wrong. Solving a paradox, invalidates the formulation of a problem proposed by the author of the paradox and leaves us with only valid procedures for solving the original problem.
Why were the previous proposed solutions for the paradox not satisfactory?
Most, if not all, the proposed solutions to Zeno’s paradox assume that Zeno’s proposed procedure is correct. The procedure seems to be logical when it is first introduced to us, but we will see that the procedure proposed by Zeno is conceptually incorrect. The authors then used a procedure similar to Zeno’s faulty procedure to reach the expected correct result for the original problem.
For example, the simple way mentioned earlier to solve the problem (not the paradox) can be examined using a spreadsheet. Given the assumptions in the diagram below, the time for Achilles to reach the tortoise is 5 seconds:
39 m / 7.8 m/s = 5 s
If we calculate a sum of an infinite series, as several mathematicians have suggested, we obtain the same result: Achilles will reach the tortoise. By applying any legitimate mathematical solution to the problem, we state, in another way that: We can prove that Achilles reaches the tortoise.
What is the problem then?
We know that Zeno is wrong: The fact that his procedure never ends, does not imply that Achilles will never reach the tortoise. We can prove this mathematically in many ways. Intuitively we all agree with the mathematicians. However, when mentally following the proposed repetitive procedure, the paradox puzzles our mind.
What are the facts?
Thus, we must conclude that Zeno’s procedure to solve the problem, is not correct.
The question is then: Why is Zeno’s procedure wrong? The key word in the previous assertions is “never”. Never implies time and the problem must be considered in the context of space and time.
The Explanation of Zeno's Paradox:
Zeno's proposition invites the solver to do a series of steps each time changing system of reference: STEP 1: The starting system of reference: The point where Achilles starts the race and the tortoise is well ahead, STEP 2: After a while, we are then asked to use a new system of reference: The point where Achilles reached and where the tortoise initially started, with the the tortoise now a bit further ahead, STEP 3: Then again we are asked to use, recursively, a new system of reference with the new starting point for Achilles and with the tortoise still further ahead, with every step we are asked to freeze the process and then continue by re-creating and examining the original problem using a different system of reference.
Today we know more about the relative motion of two bodies. Solving a problem that involves space and time, requires a defined system of reference, which cannot be changed without the proper conversions.
NOTE: The concept of a system or reference, or frame of reference, in elementary physics is founded on Einstein's Special Theory of Relativity, First Postulate: All velocities are measured relative to some frame of reference.
After Zeno's proposed first step, or first change of system of reference, the problem, as presented in the second step, is exactly the same as the original, the only change being a difference in "scale". No progress was achieved in solving the problem. Changing system of reference essentially restarts the problem-solving procedure. This realization implies that the problem is never going to reach a conclusion as the step by step procedure is reiterated. If the system of reference is changed at every step, our working spacetime shrinks with every step, the solution becomes elusive and the tortoise becomes apparently unreachable. Zeno proposes a procedure that never ends, for solving a problem that has a trivial solution.
Our Solution (Why Zeno's Formulation Is Incorrect)
Our solution of Zeno's paradox can be summarized by the following statement:
"Zeno proposes observing the race only up to a certain point, using a system of reference, and then he asks us to stop and restart observing the race using a different system of reference. This implies that the problem is now equivalent to the original and necessarily implies that the proposed procedure for solving the problem will never end."
That is, we cannot change system of reference in the middle of a problem involving velocity, space and time, whether the frame of reference is openly stated, or implied.
As an analogy, you cannot solve a problem involving measurements by using English Imperial measures at the start of calculations and then switch to metric measures (without proper conversions) in the middle of calculations.
Zeno's trick works, and puzzles our mind, because we are used to assume one frame of reference when solving this type of problems.
An example
The following is not a “solution” of the paradox, but an example showing the difference it makes, when we solve the problem without changing the system of reference. In this example, the problem is formulated as closely as possible to Zeno’s formulation.
Zeno would agree that Achilles makes longer steps than the tortoise. Let’s assume that one Achilles-step is about 20 tortoise-steps long, and let’s also assume that both Achilles and the tortoise make the same number of steps in the same amount of time. For example, two steps per second (the exact amount doesn’t really matter). If the tortoise starts the race 20 Achilles-steps ahead of him, then after 20 steps Achilles reaches where the tortoise was (See diagram below: Tortoise starting point).
In the meantime, the tortoise has made 20 of her steps, and she is now one full Achilles-step ahead of him. We have not changed our system of reference. We referred to both starting points. These did not move relatively to each other. We could choose any fixed ground point. To please Zeno, let’s continue by referring to the tortoise starting point, where Achilles currently is. When both runners make one more step, step 21, the tortoise will have moved by one of her steps and she will still be ahead of Achilles by that one tortoise-step. Achilles is now one Achilles-step ahead of the tortoise starting point. Now, let’s continue, without changing the system of reference. This is the key point. We do not redefine the problem and use the current positions of the runners as new starting points, as Zeno proposes, but we refer to the information about the race we have already accumulated in our knowledge base. Achilles then completes his 22nd step, and he is two Achilles-steps ahead of the tortoise starting point. The tortoise will have completed her 22nd tortoise-step from her starting point. Hence the tortoise is now behind Achilles by 18 tortoise-steps. Thus, if we do not change the system of reference, the paradox does not appear.
I think the assumption follows from the analysis of the motion before the catch. We see that it splits into stages.
Stage 1: Achilles is running towards the Tortoise starting point $A_1$, the Tortoise is running forward. At the moment when Achilles achieves $A_1$, Stage 1 ends and begins Stage 2. At this moment the Tortoise is at point $A_2$.
Stage 2: Achilles is running towards $A_2$, the Tortoise is running forward. At the moment when Achilles achieves $A_2$, Stage 2 ends and begins Stage 3. At this moment the Tortoise is at point $A_3$.
And so forth...
The construction of the stages show that each of them happens before the catch. This is a surprising view on a usual motion, but it sounds reasonable. So, we have that catch-up time $T_C $ is at least the sum $T_\infty=\sum T_n $ of the series of the durations of the stages $T_n$.
To show the assumption it remains to show that a strict inequality $T_C>T_\infty$ is impossible. Indeed, suppose to the contrary that after time $T_\infty$ Achilles is still behind the Tortoise. This means that a distance $d_\infty$ between them is non-zero. Let $d_0$ be the initial distance between Achilles and the Tortoise. Since Achilles is faster than the Tortoise, the $q=v_A/v_T$ of their velocities is bigger than $1$ and the distance between them decreases. But induction we can easily show that the distance $d_n$ between Achilles and the Tortoise after Stage $n$ is $d_0/q^n$. Since $q>1$, there exists $n$ such that $d_0/q^n<d_\infty$. But this is impossible, because the distance between Achilles and the Tortoise decreases and the end of Stage $n$ happens before $T_\infty$.