L02 · Real Numbers & Completeness

Interactive section

Completeness widgets (√2 gap + drag-to-build sup/inf) are on a separate page for lighter load.

Supremum

For $A\subseteq\mathbb{R}$ , $s=\sup A$ if $s$ is an upper bound and every upper bound is $\ge s$ .

Sup vs Max

$\sup A$ need not lie in $A$ . Example: $A=(0,1)$ has $\sup A=1\notin A$ .

Visualization

Open interactive Section 2

Widgets: approximating $\sqrt{2}$ in $\mathbb{Q}$ , and interactive sup/inf on the line.