(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 $b_i>a_i\ \forall i$ using induction.
Thus while $|b_i-a_i|$ tends to $0$ , we will never have $a_i>b_i$.
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 $b_i>a_i\ \forall i$, 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 <- 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