Math and science::Analysis

18 analysis theorems

The following propositions about an ordered field $$R$$ (and about associated structures such as $$[a,b]=[a,b]_R=\{x \in R:\ a \leq x \leq b\}$$) 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 $$\leq c$$, and every $$x \in B$$ is $$\geq c$$. 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 $$\epsilon > 0$$ so that every $$y$$ with $$|y-x| < \epsilon$$ 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 $$f(x) \leq B$$ 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 $$f(x) \leq f(c)$$ for all $$x$$ in $$[a,b]$$.
Does imply completeness.
8. The Mean Value Property
Suppose $$f: [a,b] \rightarrow R$$ is continuous on $$[a,b]$$ and differentiable on $$(a,b)$$. Then there exists $$c$$ in $$(a,b)$$ such that $$f'(c) = (f(b)-f(a))/(b-a)$$.
Does imply completeness.
9. The Constant Value Property
Suppose $$f: [a,b] \rightarrow R$$ is continuous on $$[a,b]$$ and differentiable on $$(a,b)$$, with $$f'(x) = 0$$ 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 $$c < 1$$, $$|f(x) - f(y)| \leq c|x-y|$$ for all $$x,y$$. Then there exists $$x$$ in $$R$$ such that $$f(x) = x$$.
Does imply completeness.
14. The Alternating Series Test
If $$a_1 \geq a_2 \geq a_3 \geq \dots$$ and $$a_n \rightarrow 0$$, then $$\sum_{n=1}^{\infty} (-1)^n a_n$$ converges.
Doesn't imply completeness.
15. The Absolute Convergence Property
If $$\sum_{n=1}^\infty |a_n|$$ converges in $$R$$, then $$\sum_{n=1}^\infty a_n$$ converges in $$R$$.
Doesn't imply completeness.
16. The Ratio Test
If $$|a_{n+1}/a_n| \rightarrow L$$ in $$R$$ as $$n \rightarrow \infty$$, with $$L < 1$$, then $$\sum_{n=1}^{\infty} a_n$$ 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?]