1. Sets

Theorem 1.1

\[\forall x \in \mathbb{R}, \exists n \in \mathbb{R}, \quad \text{where } x < n\]

Note: This does not imply that there exists $n \in \mathbb{N}$, that is greater than all real numbers!

\[\boxed{\mathbb{N \subset Z \subset Q \subset R \, (\subset C)}}\]

Interval Notation

Open Interval: $(a, b): {x \in \mathbb{R}: a < x < b}$ Closed Interval: $[a, b]: {x \in \mathbb{R}: a \leq x \leq b}$

e.g. $\mathbb{R}^+ = [0, \infty)$, as $\infty$ is NOT a real number.