Pumping Lemma Squares Proof Explanation

Question

I'm looking for some help understand this perfect squares proof using the pumping lemma. Here is the proof:

I don't understand how n^2 + k < n^2 + n towards the end of the proof. Would anyone be able to explain this to me? I also think that the last line n^2 < n^2+k < (n+1)^2 confusing because of (n^2 + k < n^2 + n).

Answer 1

First from the statement of the pumping lemma we know that $|xy|\le n$. Also we have set $k$ to be the length of $y$, i.e. $|y|=k$. Moreover the length of $xy$ is certainly greater or equal than the length of $y$: $|xy|\ge |y|$. Combining these we get

$$n\ge |xy|\ge |y|=k$$

therefore $n^2+k\le n^2+n<n^2+2n+1=(n+1)(n+1)=(n+1)^2.$

Note that equality $k=n$ may occur if and only if $x$ is empty. Otherwise $k<n$.