## New answers tagged elementary-proofs

4
votes

### Simpler proofs using the axiom of choice

The following theorem, relating path-connectedness and arc-connectedness (arc := injective path):
"Every path-connected Hausdorff space is arc-connected"
can be proven both with the axiom of ...

Community wiki

4
votes

### Simpler proofs using the axiom of choice

For an orthomorphism $A$ in a (real or complex) Banach lattice $X$ the formula $\lvert A\rvert x=\lvert Ax\rvert$ for $x\ge0$ can be obtained relatively quickly if one uses the fact that $X$ is order ...

Community wiki

7
votes

### Simpler proofs using the axiom of choice

Many examples from set theory are known, but here is a very basic (third-order) theorem from most ordinary mathematics:
"A regulated$f:[0,1]\rightarrow \mathbb{R}$ is bounded", (&)
where ...

Community wiki

21
votes

### Simpler proofs using the axiom of choice

In Division by Three, Peter G. Doyle and John Horton Conway show, without invoking the Axiom of Choice, that for any sets $A$ and $B$, if there is a bijection between $3 \times A$ and $3 \times B$, ...

Community wiki

17
votes

### Simpler proofs using the axiom of choice

Another classic example is the Schröder-Cantor-Bernstein theorem.
Theorem. If a set $A$ injects into $B$ and $B$ injects into $A$ then there is a bijection.
If AC holds, then this is nearly trivial, ...

Community wiki

14
votes

### Simpler proofs using the axiom of choice

Here is a nice example.
Theorem. Space $\newcommand\R{\mathbb{R}}\R^3$ is the disjoint union of circles.
The AC proof is simply to enumerate all points in $\R^3$ in a well-ordered sequence in length ...

Community wiki

9
votes

### Simpler proofs using the axiom of choice

There is an interesting class of problems that are provable in $ZF$ by "coincidence" in the sense that either (a fragment of) $AC$ holds and then it is provable from the proof in $ZFC$, or ...

Community wiki

7
votes

### Simpler proofs using the axiom of choice

Axiom of choice is frequently used in Analysis through the Hahn-Banach theorem. An example is the existence of solution of Dirichlet's problem (=balayage problem). This can be proved without an ...

Community wiki

4
votes

Accepted

### Analytic expression for the min value of $g(t):= \sqrt{(t-1)^2 + a^2}+ b|t|$ subject to $|t-1| \le c$

First, a few simplifications. Note that $g(-t)\ge g(t)$ and $|-t-1|\ge|t-1|$ if $t\ge0$. So, without loss of generality (wlog) $t\ge0$ and
$$g(t)=\sqrt{(t-1)^2 + a^2} + bt. \tag{1}\label{1}$$
Since $g(...

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