The questions are

Let G be the group of rational numbers under addition and n a positive integer.

a) Suppose H is a subgroup of G with index n. Show that nx is an element of H for all x in G.

I think this has something to do with nx=x+x...+x (n times) and somehow it relates to G and the fact that there is only one coset of H in G.

b) Let nG={nx element G : x element G} Show that nG=G. Hence or otherwise, prove that G has no proper subgroup of finite index.

I think if I set F to be the group of reals under addition then G is a subgroup of F. So nG can be viewed as a coset of G in F and since n is in G then nG=G (that being a property of a coset) Is that right?

But how do I show that G has no proper subgroup of finite index?