Think about it like this.
The Archimedean principle says (geometrically) that
0--------|------------------
in the below we can always find some such that
0----- -----|-------------
So, now we're claiming that is the infimum of this set. But, and so if you argue correctly we have this scenario
0---- -----------
So, we should be able to "fit something" in between zero and (by the A.P.)
but, what's the problem with that?