Most of the Zeno-like paradoxes involve an infinite sequence of events buried in the language used in the problem. Your example appears to go down the path of doubling the distance between the lines enough times to achieve the desired separation, while Zeno's original paradox depends on halving the distance.
In both cases, the tricky part is coming up with a language to describe the problem which is capable of describing this infinite series of steps without contradiction. The problematic portion is typically identifiable with an appeal to reason, such as your "... it seems we cannot create..." It is dependent on the listener agreeing with such a statement. If they agree with that statement, then typically they agree that the problem is paradoxical, and thus there must be something wrong.
If one's listener does not automatically agree with you about this statement, one must defend it. Defending it is where the really precise language often comes into play. Mathematics, for instance, has a extraordinarily precise ways of dealing with the concept of infinity, and prides itself on its self-consistency. If one phrases the problem in the language of mathematics, then one can use the strength of mathematics to argue for their position.
However, the current "preferred" solution to Zeno's paradox is calculus. Calculus handles these infinities in a way that appears to be consistent with the world we live in without causing paradoxes from self-consistency issues. If your efforts to phrase the question in mathematical terms leads you to use notations from calculus, one will find that the issue is dealt with by the handling of limits which elide away issues that might come up regarding infinitesimals. These methods have been heavily analyzed over the years, so lead people to have a great deal of confidence in answers derived from them.
In my opinion, your Zeno-like argument is going down a direction which would be proven using set theory. There are systems for handling sets like ZFC which one would be tempted to use, but your particular construction is likely to go down the path of having an infinite descending set, which is forbidden in ZFC by the axiom of regularity. One would need to look at different solutions, such as Quine atoms which are used in some non-well-founded set theories. Such set theories are much less popular than their well-founded brethren, so they do not encourage the same level of confidence.