Exercise 11.3.5

$\star$ Define a family $\mathscr{H}$ of hash functions from a finite set $U$ to a finite set $B$ to be $\epsilon$-universal if for all pairs of distinct elements $k$ and $l$ in $U$,

$$ \Pr\{h(k) = h(l)\} \le \epsilon $$.

where the probability is over the choice of the hash function $h$ drawn at random from the family $\mathscr{H}$. Show that an $\epsilon$-universal family of hash functions must have

$$ \epsilon \ge \frac{1}{|B|} - \frac{1}{|U|} $$