When I first saw a "proof" like this, the explanation of its incorrectness was something like "limits don't preserve curve length". I wasn't satisfied with that answer until I took a first course in real analysis, which explained (some of) the reasons behind calculus.

Real analysis was very satisfying, in somewhat the same way that building low-level software or libraries is satisfying -- I got to understand the guts. It was also fun to learn about erroneous historical assumptions made due to insufficient rigor. IIRC, until at least the 1870s, it was believed that any continuous function must be differentiable almost everywhere, and that in fact this should be obvious. It turns out that one can construct continuous functions that are nowhere differentiable!

This was one of my favorite topics in the class:

http://en.wikipedia.org/wiki/Construction_of_the_real_number...