Erm...I'm not sure what you're doing here. It seems to be a bit of nonsense sorry - you don't get to choose . I'll show you how to do it properly.
Claim: For a fixed , there exists a such that
Such a , if it exists, must satisfy the following inequality:
Obviously we can't actually bound for arbitrary , but we don't actually care about what happens to far away from 1. For example, we can require . Then . For such , we have
. If we want the last expression to be , we require .
To recap, we have
with subject to the conditions and . Choosing satisfies these conditions. As such you have found your desired .
Writing the proof out formally:
Fix and choose . Suppose . Then