= Solution
$A$ is uncountable. For each <binary sequence> $(\varepsilon_j)$, define
$$
f(n)=1+\sum_{j=1}^n\varepsilon_j.
$$
This is a nondecreasing function, and different binary sequences give different functions, so the uncountable set of binary sequences injects into $A$.
$B$ is countable. Every nonincreasing sequence of positive integers can decrease only finitely many times and is therefore eventually constant. Such a sequence is specified by a finite sequence of positive integers, and the set of all finite integer sequences is a <countable union of countable sets>. Hence $B$ is countable.
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