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 afixed, there exists a such that

Such a , if it exists, must satisfy the following inequality:

Now

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

since

since