Would someone please give me some help with my case 2 below?

The problem: Let x 0, x such that for any > 0, x . Prove that x = 0

Solution:

Case 1: Let x > 0 and =

then

= > 0 because x > 0

since > 0 by given, x

Case 2: Let x = 0 and =

then

= 0/2 = 0

since = 0 = x, I cannot conclude x

unless I let = x + y (where y ). But can I do that? Some thing does not look right in my opinion, but I am stuck and unsure.

Thanks,