Let $X$ be the set of those sequences $s \in \mathbb N^ \mathbb N$, in which appear all of the natural elements (for any $k$ exists $n$, $s(n)=k$). We define $\leqslant$ order on set $X$, $s \leqslant k$ if and only if $s(n) \leqslant k(n)$ for any $n$. Does $(X, \leqslant)$ have the smallest element? Does it have a minimal element?
My attempts: If i understand the idea of minimal and smallest elements well, this set has no minimal or smallest elements. For every sequence we can find a smaller one. Is that correct?