Simple statements to prove or disprove

Simple statements to prove or disprove

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, forall $n \in \Bbb Z$, one has $n
\leq m$.
It would be great to cover different type of statements.