How would you prove that is presentation of
what is you DEFINITION of the Prüfer group?
the commutator relation [xi,xj] = 1, expresses the fact that your group is abelian, the relation (xi+1)p = xi expresses the fact that every element of <xj> is a pj-th root of unity.
it should be clear that we have a tower of inclusions:
<x1> ⊂ <x2> ⊂ <x3> ⊂ ......
where <xj> ≅ Z/pj under addition modulo pj.