This map, restricted to the subset of

consisting of elements of the form

, is an injection; moreover when

, we have

; hence there are infinitely many points of

in the bounded set

. Immediately from the Bolzano-Weierstrass theorem we know that

has an accumulation point. It is then an easy matter to show that any additive subgroup of

having an accumulation point is dense in

. To see this, suppose you have an open interval

of length

in

. Then by the above there is an element of

with 0<|s|<L; taking all its multiples (which are all elements of

) we see that one of them must fall in

.