How would you prove that is presentation of
what is you DEFINITION of the Prüfer group?
the commutator relation [x_{i},x_{j}] = 1, expresses the fact that your group is abelian, the relation (x_{i+1})^{p} = x_{i} expresses the fact that every element of <x_{j}> is a p^{j}-th root of unity.
it should be clear that we have a tower of inclusions:
<x_{1}> ⊂ <x_{2}> ⊂ <x_{3}> ⊂ ......
where <x_{j}> ≅ Z/p^{j} under addition modulo p^{j}.