Yeah, it is not entirely obvious. You need to prove that if you fix a number x from range [0,1] then the point at "x * length(Hn)" converges to a point in 2D, for every x, as n goes to infinity (Hn = nth Hilbert curve).

The limit for x is the mapping from x to a point in 2D.

The proof is by observing that the subsequent "jumps" are exponentially smaller.