Type up the following proofs in LaTeX. Use the semiformal proof style we have been using, one statement per line, and a reason stated for each line that needs one, with optional comments where you feel they would help the reader. You can do that by clicking on the Homework Template link below on this page. To hand in your document, download the pdf and put your pdf file in your Dropbox folder with the appropriate filename. All variables and constants are assumed to be real numbers unless stated otherwise.

You can use any theorems prior to the chapter on real numbers (keeping in mind that theorems about the natural numbers do not automatically extend to theorems about the reals). Other than that you can only use the axioms and definitions about the real numbers, and any theorem you prove first to use in a later proof (of course). But you cannot use any theorems in the problem sections in Chapter 9 in the Lecture Notes.

- 0.
**A warm up**$\negative{1}\neq 1$. - 1.
**One is positive**$0<1$ - 2.
**Double negative**For all real numbers $x$, $$\negative{\left(\negative{x}\right)}=x$$ - 3.
**Alternate additive inverse**$\negative{x}=\negative{1}\cdot x$ - 4.
**Signed products**$\negative{x}\cdot\negative{y}=x\cdot y$ - 5.
**Alternate multiplicative inverse**$x^-=\frac{1}{x}$ - 6.
**Trichotomy**For any real numbers $x,y$ exactly one of the following is true:- $x\lt y$
- $x=y$
- $y\lt x$

- 7.
**Product of negatives is positive**$\negative{x}\cdot\negative{y}=x\cdot y$ - 8.
**Some reals are natural**Consider the sequence of real numbers, $(N_k)_{k=0}^\infty$, defined at the beginning of section 9.4 in the notes. All of the Peano axioms hold for $(N_k)_{k=0}^\infty$. Let’s check a few.- Show Peano Axiom N3 holds for $(N_k)_{k=0}^\infty$.
- Show Peano Axiom M1 holds for $(N_k)_{k=0}^\infty$.
- Show Peano Axiom I holds for $(N_k)_{k=0}^\infty$.