# Lecture - Analysis MT22, IV

> Source: https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/lectures/4/ · Updated: 2022-11-01 · Tags: uni, lecture

- [Course - Analysis MT22](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/)

### Flashcards
What is the approximation property for $\sup S$?::
$$
\forall \epsilon > 0\\,\exists x\in S \text{ s.t. } \sup S - \epsilon < x \le \sup S
$$

What is the approximation property for $\inf S$?::
$$
\forall \epsilon > 0\\, \exists x\in S \text{ s.t. } \inf S \le x < \inf S + \epsilon 
$$

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 $\epsilon$?::
$$
\forall \epsilon > 0\\,\exists n \in \mathbb{N} \text{ s.t. } 0<\frac{1}{n}<\epsilon
$$

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.

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