You're right, that's where you have a problem. is increasing, hence is one-to-one, but it not onto, so you can't define for any .

You can patch this by saying: let us define the "generalized inverse function" by . By doing so, you still can say:

and it works.3. We know because converges that for every there exists and integer such that .

4. So from this we can make the observation that whenever .

But honestly, it seems to me that this is exactly the usual proof, written a bit differently... (with an explicit )