  We develop a new multi-scale framework flexible enough to solve a number of
problems involving embedding random sequences into random sequences. Grimmett,
Liggett and Richthammer asked whether there exists an increasing M-Lipschitz
embedding from one i.i.d. Bernoulli sequences into an independent copy with
positive probability. We give a positive answer for large enough M. A closely
related problem is to show that two independent Poisson processes on R are
roughly isometric (or quasi-isometric). Our approach also applies in this case
answering a conjecture of Szegedy and of Peled. Our theorem also gives a new
proof to Winkler's compatible sequences problem.
