Prove that the language $\{ww \mid w \in \{a,b\}^*\}$ is not FA (Finite Automata) recognisable. Hint: Assume that $|xy| \le k$ in the pumping lemma.
I have no idea where to begin for this. Any help would be much appreciated.

It's also possible — and perhaps simpler — to prove this directly using the pigeonhole principle without invoking the pumping lemma.
Namely, assume that the language $L = \{ww \,|\, w \in \{a,b\}^*\}$ is recognized by a finite state automaton with $n$ states, and consider the set $W = \{a,b\}^k \subset \{a,b\}^*$ of words of length $k$, where $2^k > n$.
By the pigeonhole principle, since $|W| = 2^k > n$, there must be two distinct words $w,w' \in W$ such that, after reading the word $w$, the automaton is in the same state as after reading $w'$.
But this means that, since the automaton accepts $ww \in L$, it must also accept $w'w \notin L$, which leads to a contradiction.
