Strange question

Yes, this is a little unclear.

This may be a little harsh, but I flagged the presented solution as GBS, for the following reasons: It should be cristal clear that an infimum has to satisfy two conditions, being less than any number of the set, and being the largest number with this property. This reasoning is somewhat there, but too convoluted in my opinion. Also, I would like to see some intuition applied here. And a clear plan of attack. Why y is not allowed to be positive is magic and only become clear later on. In fact, some more intuition rather than pure mathematical automatism applied here would make it easy to argue that we just have to check for candidates ${\sqrt {5}}<z<0$ in the first place.

I rewrote the proof completely using your feedback. We can still find the previous proof in the history of the page if needed. And as such, I'm flagging as RS again.

Well done, much clearer now. QGS.