Lecture - Analysis MT22, IV
Flashcards
What is the approximation property for $\sup S$?
What is the approximation property for $\inf S$?
What does $\forall \epsilon > 0\,\exists x\in S \text{ s.t. } \sup S - \epsilon < x \le \sup S$ say in English?
Anything even a tiny bit less than the supremum is less than an $x$ in $S$.
What is the simpler statement of the Archimedean property?
$\mathbb{N}$ is not bounded above.
What is the more complicated statement of the Archimedean property, involving $\epsilon$?
How do you prove the more basic statement of the Archimedean property, that $\mathbb{N}$ is not bounded above?
Assume $\mathbb{N}$ is bounded above and then use the approximation property with $\epsilon = 1$.
How do you prove the more complicated statement of the Archimedean property, that $\forall \epsilon > 0\,\exists n \in \mathbb{N} \text{ s.t. } 0<\frac{1}{n}<\epsilon$?
Assume the opposite for a contradiction and then use the fact that $\mathbb{N}$ is not bounded above.