Can someone explain the relation between "Achilles chasing turtle" paradox and monotonic, bounded sequence to me?

Question

Recently I have read a book called "Caculus - Basic Concepts for High School"

The possibility of infinite but bounded sets was not known, for example, to ancient Greeks. Suffice it to recall the famous paradox about Achilles chasing a turtle. Let us assume that Achilles and the turtle are initially separated by a distance of 1 km. Achilles moves 10 times faster than the turtle. Ancient Greeks reasoned like this: during the time Achilles covers 1 km the turtle covers 100 m. By the time Achilles has covered these 100 m, the turtle will have made another 10 m, and before Achilles has covered these 10 m, the turtle will have made 1 m more, and so on. Out of these considerations a paradoxical conclusion was derived that Achilles could never catch up with the turtle. This "paradox" shows that ancient Greeks failed to grasp the fact that a monotonic sequence may be bounded.

I know how to prove the conclusion of this paradox is incorrect, and understand that the described movement of Achilles and the turtle each is a monotonic, bounded sequence. However, I don't understand author's claim that

This "paradox" shows that ancient Greeks failed to grasp the fact that a monotonic sequence may be bounded.

Can someone help to explain this part? Thank you very much.

EDIT: Thank you all so much for your instant replies. However, I still don't get how these two movements being strictly monotonic, bounded sequences, has anything to do with Greek ancient people's conclusion.

For Achilles' movement: $$ x = \sum_{i=0}^\infty \frac{1000}{10^i}$$

For the turtle's movement: $$ y = 1000 + \sum_{i=0}^\infty \frac{100}{10^i} $$

So the Greek ancient people are trying to find an $n$ where: $$ x_n = y_n $$

However, since $ y_n = x_{n+1} $ and $ x_n < x_{n+1} $ so there is no such $n$ exist.

With everything mentioned above, I still don't find anything to do with bounded sequence part.

Answer 1

I think it means that they did not know that there could be sequences where every value is bigger than the last that themselves have a limit, for example the sequence: 1/2, 2/3, 3/4, 4,5, 5/6, etc it has a limit of 1 but it always increses, the statement says that the greek did not have the understanding that there where such sequences, where the numbers always increse never get to infinity.

Answer 2

(For fun)
Greek philosopher:
Now you see, by the time Achille reaches the place where the turtle is right row, the turtle will have gone to another place. So Achill will have to run an extra time to reach turtle again. Clearly, that pattern repeats infinitely to Achille, thus Achille has to run forever.
Greek student
But Philosopher, why does AChille have to run forever?
Greek Philosopher:
Stupid! Because all the instants -at which that pattern happens to Achille - is an infinite increasing sequence. And an infinite increasing sequence cannot be bounded.
Count your cows! Count the houses! Do you see that even though those numbers may be big but you still can count them all. That is because they are bounded. Finite is bounded! Infinite is very very big!
You potato head surely have wasted all the potatoes you have eaten.