This question is related to this one: Continued fractions using all natural integers. Suppose we have the set of natural numbers $N$ with order and we perform permutation on it. So we obtain the same elements with different order. Suppose we describe such permutations by usual notation when (1,2,3,4,5...) means identity permutation. Then lets say that permutation denoted by (1,3,2,4,5,6...) ( from the 4th place there is list of natural numbers in usual order) is finite because it only mixes numbers 1,2,3 -> 1,3,2 and for remaining elements it is identity permutation. As I find here there is definition of such objects, namely a few possibilities as states the answer of Qiaochu Yuan.

Questions:

1. Are infinite permutation decomposable into cycles? Transpositions?
2. Is possible to find such permutation of natural numbers that it cannot be a limit of finite permutations?

This question is related to this one: Continued fractions using all natural integers. Suppose we have the set of natural numbers $N$ with order and we perform permutation on it. So we obtain the same elements with different order. Suppose we describe such permutations by usual notation when (1,2,3,4,5...) means identity permutation. Then lets say that permutation denoted by (1,3,2,4,5,6...) ( from the 4th place there is list of natural numbers in usual order) is finite because it only mixes numbers 1,2,3 -> 1,3,2 and for remaining elements it is identity permutation. As I find here there is definition of such objects, namely a few possibilities as states the answer of Qiaochu Yuan.

Questions:

1. Are infinite permutation decomposable into cycles? Transpositions?
2. Is possible to find such permutation of natural numbers that it cannot be a limit of finite permutations?