# Why natural numbers don't have supremum?

As I know the definition of supremum for set $$S$$ is the lowest number that is greater or equal than all the members of $$S$$. This means : $$\forall \: m\in R \:\:m<\sup S,\exists\: s\in S \rightarrow s> x$$. Based on this definition we have supremum for all finite subset of $$\mathbb N$$. For example the $$\sup S$$ for $$S = \{1,2,\dots, k\}$$ is $$k$$.

Is this right?

• The supremum of any finite set is it maximum Nov 2, 2020 at 12:58
• @MauroALLEGRANZA $\mathbb R$ is also infinite, why it has $\sup$? Nov 2, 2020 at 13:00
• $\mathbb R$ has no sup. You can prove that by contradiction easily. Nov 2, 2020 at 13:00
• Then what is the definition of a complete set? as I know field F is complere if every upper bounded subset of F has $\sup$ , this condition is correct for natural numbers too but they're not complete. Nov 2, 2020 at 13:06
• The set of natural numbers is not a field !
– Fred
Nov 2, 2020 at 13:11

Let's review several senses in which $$\Bbb R$$ is complete:
• Each set with an upper bound has a least upper bound This works in $$\Bbb N$$ because $$\Bbb N\subseteq\Bbb R$$, so if $$S\subseteq\Bbb N$$ then $$S\subseteq\Bbb R$$. Narrowly interpreted, your question boils down to overlooking the bold part; broadly interpreted viz. the comments, other respects in which $$\Bbb R$$ is complete are worth comparing with $$\Bbb N$$.
• Each Cauchy sequence converges to a value in $$\Bbb R$$ Any Cauchy sequence in $$\Bbb N$$ is eventually constant, converging to some natural number.
• Intersection of interval nested, in the way the nested intervals theorem specifies, is a $$1$$-element set Since naturals differ by at least $$1$$ (crucial also in the below discussion of IVT), you eventually can't nest proper-subset intervals further.
• Monotone convergence In analogy with the least-upper-bound point above, this follows because every sequence in $$\Bbb N$$ is also in $$\Bbb R$$. Bolzano-Weierstrass Ditto.
• Intermediate value theorem This fails for $$\Bbb N$$ because if $$f(a)<0 then there might not be values between $$a,\,b$$.