18 analysis theorems

The following propositions about an ordered field $R$ (and about associated structures such as $[a,b]=[a,b{]}_R=\{x\in R:a\le x\le b\}$) are true when the ordered field $R$ is taken to be $\mathbb{R}$, the field of real numbers.

1. The Dedekind Completeness Property (Supremum)

Suppose $S$ is a nonempty subset of $R$ that is bounded above. Then there exists a number $c$ that is an upper bound of $S$ such that every upper bound of $S$ is greater than or equal to $c$.
We refer to this as completeness.

2. The Archimedean Property

For every $x>0,x\in R$ there exists $n\in \mathbb{N}$ with $n>x$. Equivalently, for every $x>0,x\in R$ there exists $n\in \mathbb{N}$ with $1/n<x$.
Doesn't imply completeness.

3. The Cut Property

Suppose $A$ and $B$ are nonempty disjoint subsets of $R$ whose union is all of $R$, such that every element of $A$ is less than every element of $B$. Then there exists a cutpoint $c\in R$ such that every $x<c$ is in $A$ and every $x>c$ is in $B$. (Or, if you prefer: Every $x\in A$ is $\le c$, and every $x\in B$ is $\ge c$. It's easy to check that the two versions are equivalent.) Since this property may be unfamiliar, we remark that the Cut Property follows immediately from Dedekind completeness (take $c$ to be the least upper bound of $A$).
Does imply completeness.

4. Topological Connectedness

Say $S\subseteq R$ is open if for every $x$ in $S$ there exists $ϵ>0$ so that every $y$ with $|y-x|<ϵ$ is also in $S$. Then there is no way to express $R$ as a union of two disjoint nonempty open sets. That is, if $R=A\cup B$ with $A,B$ nonempty and open, then $A\cap B$ is nonempty.
Does imply completeness.

5. The Intermediate Value Property

If $f$ is a continuous function from $[a,b]$ to $R$, with $f(a)<0$ and $f(b)>0$, then there exists $c$ in $(a,b)$ with $f(c)=0$.
Does imply completeness.

6. The Bounded Value Property

If $f$ is a continuous function from $[a,b]$ to $R$, then there exists $B$ in $R$ with $f(x)\le B$ for all $x$ in $[a,b]$.
Doesn't imply completeness.

7. The Extreme Value Property

If $f$ is a continuous function from $[a,b]$ to $R$, then there exists $c$ in $[a,b]$ with $f(x)\le f(c)$ for all $x$ in $[a,b]$.
Does imply completeness.

8. The Mean Value Property

Suppose $f:[a,b]\to R$ is continuous on $[a,b]$ and differentiable on $(a,b)$. Then there exists $c$ in $(a,b)$ such that $f^{\prime }(c)=(f(b)-f(a))/(b-a)$.
Does imply completeness.

9. The Constant Value Property

Suppose $f:[a,b]\to R$ is continuous on $[a,b]$ and differentiable on $(a,b)$, with $f^{\prime }(x)=0$ for all $x$ in $(a,b)$. Then $f(x)$ is constant on $[a,b]$.
Does imply completeness.

10. The Convergence of Bounded Monotone Sequences

Every monotone increasing (or decreasing) sequence in $R$ that is bounded converges to some limit.
Does imply completeness.

11. The Convergence of Cauchy Sequences

Every Cauchy sequence in $R$ is convergent.
Doesn't imply completeness.

12. The Fixed Point Property for Closed Bounded Intervals

Suppose $f$ is a continuous map from $[a,b]\subset R$ to itself. Then there exists $x$ in $[a,b]$ such that $f(x)=x$.
Does imply completeness.

13. The Contraction Map Property

Suppose $f$ is a map from $R$ to itself such that for some constant $c<1$, $|f(x)-f(y)|\le c|x-y|$ for all $x,y$. Then there exists $x$ in $R$ such that $f(x)=x$.
Does imply completeness.

14. The Alternating Series Test

If $a_1\ge a_2\ge a_3\ge \dots$ and $a_n\to 0$, then $\sum _{n=1}^{\mathrm{\infty }}(-1{)}^n{a}_n$ converges.
Doesn't imply completeness.

15. The Absolute Convergence Property

If $\sum _{n=1}^{\mathrm{\infty }}|a_n|$ converges in $R$, then $\sum _{n=1}^{\mathrm{\infty }}{a}_n$ converges in $R$.
Doesn't imply completeness.

16. The Ratio Test

If $|a_{n+1}/a_n|\to L$ in $R$ as $n\to \mathrm{\infty }$, with $L<1$, then $\sum _{n=1}^{\mathrm{\infty }}{a}_n$ converges in $R$.
Does imply completeness.

17. The Shrinking Interval Property

Suppose $I_1\supseteq I_2\supseteq \dots$ are [...] in $R$ with [...]. Then [...].
[Does/doesn't imply completeness?]

18. The Nested Interval Property

Suppose $I_1\supseteq I_2\supseteq \dots$ are [...]. Then [...]. (Another name for this property is "spherical completeness''.)
[Does/doesn't imply completeness?]