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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?

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Yes, that is correct (and lack of minimal elements immediately implies lack of a smallest, or minimum, element as well). But you need to justify your answer. Just claiming something doesn't make it right. Here is the key idea:

Given any sequence, since it has non-zero elements, you can always "stretch" the sequence and add $0$'s in the new places, thus making the new sequence smaller.

For example, $s(n)=n$ is not minimal since taking $k(0)=k(1)=0$ and $k(n)=n-1$ for $n>1$ is strictly smaller.

Formalizing this could be a bit of a pain in the lower lower-back, but it is a good exercise in really understanding why there are no minimal elements.

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If $s\in X$, define$$s'(n)=\begin{cases}1&\text{ if }n=1\\s(n)-1&\text{ otherwise.}\end{cases}$$Then $s'<s$. Besides, $s'\in X$:

  • Since $s\in X$, there is a $k\in\mathbb N$ such that $s(k)=1$. Therefore, $s'(k)=1$.
  • Take $n\in\mathbb{N}\setminus\{1\}$. Since $s\in X$, there is a $k\in\mathbb N$ such that $s(k)=n+1$. Therefore, $s'(k)=n$.

So, $X$ has no smallest element.

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  • $\begingroup$ You need to argue why $s'\in X$ as well. Don't forget that. $\endgroup$ – Asaf Karagila Aug 11 '17 at 11:08
  • $\begingroup$ @AsafKaragila Done. $\endgroup$ – José Carlos Santos Aug 11 '17 at 11:12
  • $\begingroup$ Now only accept the inevitable truth that $0\in\Bbb N$, and you should be fine... :D $\endgroup$ – Asaf Karagila Aug 11 '17 at 11:12
  • $\begingroup$ @AsafKaragila Not even under torture. ;-) $\endgroup$ – José Carlos Santos Aug 11 '17 at 11:13
  • $\begingroup$ You forget that set theorists have methods of forcing you to admit to certain truths! ;-) $\endgroup$ – Asaf Karagila Aug 11 '17 at 11:14

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