$\newcommand{\card}{\operatorname{card}}$
Let $A$ be a nonempty countable set of real numbers, and $0< a\leq b$. Is the following true:
$$ \tag{*} \inf_{x\in \mathbb R} \card(A\cap [x,x+a]) \geq \inf_{x\in \mathbb R} \card(A\cap [x,x+b]) $$
where $\card$ means number of elements in the set.
As far as I know, since $a\leq b$ then $$A\cap [x,x+a] \; \subseteq \; A\cap [x,x+b] $$
so $$ \card(A\cap [x,x+a]) \; \leq\; \card(A\cap [x,x+b]) $$ for all $x\in \mathbb R$. How I can complete the proof (if the result is correct)?
(I know that if we have two sets $B\subseteq C$ then $\inf B \geq\inf C$. But how I can use this in terms of cardinality of sets?)