OK, some critique. One of the main rules is that every variable must be introduced either by "for all", "any", etc. or "there exists", "some", etc.

The structure of you sentence is as follows: For any a and b ..., where ..., such that ... and ... . What happens for any such a and b? Next, "where k is an integer (which relies on a and b)" is not good because it is not clear how precisely k relies on a and b. Finally, you did not introduce a_2, a_3, etc.

I would say: "for any , define if there exist a sequence of elements of where and for all ".

Now that the definition has ended, the scope of a, b, k, a_1, ..., a_k is closed and they are just names with no assigned meanings. The proof opens a new scope. So, to answer your question, it does not make sense to choose a_2 as an arbitrary element of the sequence between a and b. So far, a and b are undefined. You should say, e.g.: Consider arbitrary and assume that and .

I think a remark about the relationship between and for various would be merited since the actual definition does not mention powers of . This relationship may seem obvious to a sophisticated reader, but our discussion is not at that level.