"$n$ is even iff $n^2$ is even" and other simple statements to teach proof-writing I am supposed to teach undergraduate students who do not major in mathematics and I would like to give them a short introduction to mathematical reasoning and to the concept of proof. I am looking for very simple mathematical statements (true or false) to help them get familiar with logic and proof writing.
I am not sure if what I ask is clear, so here are some ideas I came up with:


*

*Let $n$ be an integer. If $n$ is a multiple of $42$, then $n$ is even.

*Let $n$ be an integer. If $n$ is a multiple of $43$, then $n$ is odd.

*If $x$ is a positive real number, then $x^2 \geq x$.

*For all $n \in \Bbb Z$, $n$ is even iff $n^2$ is even.

*There is no $x \in \Bbb Q$ such that $x^2 = 2$.

*There exist irrational numbers $\alpha,\beta > 0$ such that $\alpha^\beta$ is rational.

*For all $x \in \Bbb R$, if [for all $\epsilon > 0$, $|x| < \epsilon$], then $x = 0$.

*There exists $m \in \Bbb Z$ such that, for all $n \in \Bbb Z$, one has $n \leq m$.

*$(-1)\times (-1) = 1$

*For all $x \in \Bbb R$, one has $0\times x = 0$. (proposed by Robert Auffarth)

*Let $n$ be a positive integer. If $x$ is real and $x^n=0$ then $x = 0$. (ibidem)

*Suppose $b,d \neq 0$ and $\dfrac{a}{b} = \dfrac{c}{d}$. Then $\dfrac{a}{b} + \dfrac{c}{d} = \dfrac{a+c}{b+d}$.
It would be great to cover different type of statements.
 A: *

*There are an infinite number of primes (i.e., the proof by Euclid).

*$a^p\equiv a\mod p$ implies $p$ is prime.
A: (Related to your #12)
Let $a,b,c,d$ be positive integers.  Prove that if $\frac{a}{b}<\frac{c}{d}$ then
$\frac{a}{b}<\frac{a+c}{b+d}<\frac{c}{d}$.  (Similarly for $>$ and $=$.)
A: An interesting example of a statement they could try to prove is:

Let $k \in \mathbb{Z}$. If $2k$ is odd, then $k$ must be odd.

This seems to be made for contradiction or contrapositive, which work well. However, while it is a true statement, it could be improved. This could lead to a useful discussion about strengthening hypotheses/conclusions, the difference between necessary and sufficient conditions, and vacuous statements. Some of these things may be too advanced for the students you're teaching, but I think the example is simple enough for them to understand if you want to challenge them.
A: This may be too hard for non-math students: Any real polynomial of degree $n$ has at most $n$ roots.
Other fun ones that students know but don't know how to prove are: 


*

*Prove that $0\cdot x=0$ for any real number $x$.

*Prove that $1\cdot x=x$ for any real number $x$.

*If $x^n=0$ for a real number $x$, then $x=0$.
A: For each $\epsilon >0$, there exists $c$ such that for each $x > c $, $\left| \dfrac{3x-1}{2x+11} -\dfrac{3}{2}\right| < \epsilon$.
