and since the LHS goes to 0 (convergence in Lē), then the RHS goes to 0 too, and hence the convergence in measure.
I'm rephrasing it
Since converges to in Lē, then by definition of the convergence in Lē, the red part goes to 0 as n goes to infinity
Thus by the sandwich theorem, the blue part goes to 0 as n goes to infinity.
When it is said "for all epsilon", it's like you choose any epsilon and you fix it. So it's like a constant !
Thus , which means, by definition, that converges to in measure.
Looks better ?