paradoxical proof by induction

Eubulides was one of Euclides students.Euclides is the "father" of geometry you have studied so far.
Well, Eubulides was pretty famous for his logical paradoxes one of which is :
"this statement is false".
Another example (modified) is the following paradoxical proof by induction :
 Every positive integer is a whole lot less than 1000000.

Proof: 
Base case. 1 is certainly a whole lot less than 1000000.

Induction step. 
If the positive integer k is a whole lot less  than  1000000, then certainly k+1 , which is just slightly bigger than  k, is still a whole lot less than  1000000.

Hence, by mathematical induction, the statement is true for all positive integers.
In particular, a billion is a whole lot less than a million.

What would you say?