i´ve been studying an article from W. P. Thurston about hyperbolic geometry, there, he defines something called left earthquake, whose definition is as follows:
Definition. If $\lambda$ is a geodesic lamination on a hyperbolic plane, a $\lambda$-left earthquake map $E$ is a (possibly discontinuous) biyective map from the hyperbolic plane onto itself which is an isometry on each stratum of $\lambda$. Furthermore, the map $E$ satisfies the condition that for any two strata $A\neq B$ of $\lambda$, the comparison isometry $$\operatorname{cmp}(A,B)=(E\mid A)^{-1}\circ (E\mid B) : \mathbb{H}^2\to\mathbb{H}^2$$ is a hyperbolic transformation whose axis weakly separates $A$ and $B$ and which translates to the left as viewed from $A$.
Of course, it is easily seen that any right earthquake induces a left earthquake and viceversa. Then, Thurston establishes the following:
Theorem. Left earthquake maps with finite laminations are dense in the set of all left earthquake maps, in the topology of uniform convergence on compact sets.
He starts the proof by first noticing that the image of a compact set by a left earthquake map has a bounded diameter. Of course, I believe that the proof relies on the fact that compact sets on a metric space are bounded, and thus, under a continuous function, the image of a compact set is bounded, i.e., has bounded diameter. (Thurston argues that the proof of this fact is in analogy to the proof of the following: The image of a compact interval under a monotone function is bounded, but I´m not able to figure out a proof of the latter).
Thurston then gives the following argument:
Blockquote Given any left $\lambda$-earthquake $E$, then for any compact subset $K$ of the hyperbolic plane, there is a finite subset of strata which intersect $K$ such that the union of the images of the images of this finite set of strata in the graph of $E\subseteq\mathbb{H}^2\times\mathbb{H}^2$ is $\varepsilon$-dense in the graph of $E$ restricted to $K$.
This argument is not very clear to me, any help with trying to understand this would be very helpful
