Question 62
Let $p,q$ be positive integers. For every $a,b\in\mathbb R$, define
$$
P(a)=\left\{a_n=a+\frac np\;\middle|\;n\in\mathbb N\right\},
\qquad
Q(b)=\left\{b_n=b+\frac nq\;\middle|\;n\in\mathbb N\right\}.
$$
The distance from $P(a)$ to $Q(b)$ is defined as the minimum of $|x-y|$ over $x\in P(a)$ and $y\in Q(b)$.
Determine the largest possible distance between $P(a)$ and $Q(b)$ as $a$ and $b$ vary over $\mathbb R$.
$$
P(a)=\left\{a_n=a+\frac np\;\middle|\;n\in\mathbb N\right\},
\qquad
Q(b)=\left\{b_n=b+\frac nq\;\middle|\;n\in\mathbb N\right\}.
$$
The distance from $P(a)$ to $Q(b)$ is defined as the minimum of $|x-y|$ over $x\in P(a)$ and $y\in Q(b)$.
Determine the largest possible distance between $P(a)$ and $Q(b)$ as $a$ and $b$ vary over $\mathbb R$.