Zeno's paradox cannot be disproven with algebra or any math. Why? Because it is not a mathematical paradox.
Obviously, Zeno knew that Achilles can catch up with the tortoise. Zeno was not trying to prove that Achilles cannot catch up with the tortoise, or that a runner cannot run around a stadium.
Zeno's point was to illustrate that just because something is logical does not necessarily mean it is true. The flaw is not in logic as such. The flaw is in human understanding (or lack thereof) and applying logic to premises that appear correct.
In the case of the runner the "logic" was that the runner cannot run around the stadium because first he has to run around half of the stadium, then he has to run around half of the remaining distance, then around half of the remaining distance, and so on, ad infinitum.
It is irrelevant that our current knowledge of mathematics can easily disprove it. At Zeno's time it appeared correct. The premise was false but was perceived as true.
At our own time some other premise may appear true and still be false, simply because our knowledge and understanding are limited.
What Zeno's paradox teaches us today, as it did in Zeno's time, is that we should not be arrogant about our knowledge and understanding, that we should validate all theory by practical observation, and that we should always be ready to admit that we can be wrong and often are. In other words, we should be ready to change our theory no matter how feasible and logical it may sound, and how convinced we may be about its validity.
Zeno's paradox is not about mathematics, it is about keeping an open mind. And the need for that cannot be disproved.