Zeno's Achilles & Tortoise - Where exactly is the proof wrong? (For those who don't know what this paradox is see Wikipedia  or the Stanford Encyclopedia of Philosophy.)
Let us define $a_i$ and $b_i$ recursively
$$
a_0 = 0\\
b_0 = 1\\
a_i = a_{i-1} + (b_{i-1} - a_{i-1})\\
b_i = b_{i-1} + (b_{i-1} - a_{i-1})/2
$$
It is easy to prove that $b_i&gt;a_i\  \forall i$ using induction.
Thus while $|b_i-a_i|$ tends to $0$ , we will never have $a_i&gt;b_i$.
We can now just replace $a_0$ as Achilles start position and $b_0$ as Tortoise start position. And then subsequent positions of Achilles is given by $a_i$s (Achilles new position is = Tortoise old position, which is the $1^{st}$ recursion). And Tortoise is assumed to move at half the speed of Achilles. Tortoise positions are represented by $b_i$s. (So, new position of Tortoise = Old Position + 1/2 the distance traveled by Achilles, which is the $2^{nd}$ recursion.)
Given, we have proven $b_i&gt;a_i\  \forall i$, thus I claim Achilles will always be behind Tortoise (He will come closer and closer but will never overtake).
Obviously, I'm wrong but exactly where / which step of the proof above? (Please provide the exact mathematical step/argument where I went wrong.)
Some further discussion:
Basis the responses I got (which I'm unable to find fully convincing - and it maybe just me that I don't understand them well enough) I would like to add - In my opinion, the way I have defined $a_i$ and $b_i$ it is just a subset of positions that Achilles and Tortoise can take. In that subset what I have proved is correct i.e. Achilles cannot overtake Tortoise. But just in that subset &lt;- And I think this is the key
Note that my $a_i$ and $b_i$ are all rational. I can embed infinite rationals between any 2 points on the real line. I think fundamentally the error in my proof is that I use induction on continuous variables. I'm not formally trained to express that mathematically in a precise way - Hence this question.
My question is not to challenge/discuss that Achilles will overtake or not etc or to come-up with another proof - My precise question is where exactly is my proof wrong.
Thanks
The issue is as follows. You have constructed an infinite sequence of times, at all of which Achilles is behind the tortoise. However, that doesn't mean that Achilles will always be behind the tortoise, because the set of times you have constructed is bounded.  Suppose Achilles has unit speed. Then they reach positions $a_1$ and $b_1$ at time $1$, $a_2$ and $b_2$ at time $3/2$, $a_3$ and $b_3$ at time $7/4$, and so on. It is easy to verify that all these times are less than $2$, so your argument only implies the tortoise is ahead for $t&lt;2$. (Indeed, $t=2$ is exactly when Achilles overtakes the tortoise.)
There is nothing wrong with your proof… internal to the proof.
In respect of the universe of consideration, what is wrong with your proof is that it adopts a constrained frame of reference.  It declines to consider any value outside of the constraint that Achilles &lt; tortoise.
In other words… as a proof… it proves that Achilles will always be behind the tortoise, given the assumption that Achilles will always be behind the tortoise.
Mathematically, the idea is to follow the approach in Sameer Baheti’s answer [noting that I have not checked the actual maths] — you allow yourself to use a different frame of reference — one that is not artificially constrained.