Density of sets of natural numbers and the Levy group

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Let denote the set of positive integers. The asymptotic density of the set A is d(A) = ₙ→∞ |A∩ [1,n]|/n, if this limit exists. Let AD denote the set of all sets of positive integers that have asymptotic density, and let S denote the set of all permutations of the positive integers. The group L consists of all permutations f ∈ S such that A ∈ AD if and only if f(A) ∈ AD, and the group L consists of all permutations f ∈ L such that d(f(A)) = d(A) for all A ∈ AD. Let f: → be a one-to-one function such that d(f())=1 and, if A ∈ AD, then f(A) ∈ AD. It is proved that f must also preserve density, that is, d(f(A)) = d(A) for all A ∈ AD. Thus, the groups L and L coincide.