Sep 02⌗

A subset \(S\) of \(\R\) is dense if and only if \(\forall a,b \in \R, a \lt b, \exists x \in S \ni a \lt x \lt b\)

Theorem 1. \(\mathbb{Q}\) is a dense subset of \(\R\)

\(a \lt \frac{m}{n} \lt b\)

\(na \lt m \lt nb\)

\(nb-na \gt 1 \Rightarrow a-b \gt \frac{1}{n}\)

By a thm, \(\exists m \in \Z \ni nb-1 \le m \lt nb\)

Let a < b be two reals since b-a > 0 by AP ∃ n ∈ \N ∋ 1\m < b-a implies 1 < nb - na ⇒ na > nb-1

For irrational proof, use addition

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Let \(\epsilon\) be positive By AP there is a natural number N st. \(1/N \lt \epsilon\) If \(n \ge N\) then \(\abs{1\n - 0} = 1\n \le 1\N \le epislon\)

Basic strucut - epislon positive, ap shows some big N less than episolon number exists, If n ≥ N, then start with abs and redo the simplicifaction to show all steps anre great than before, and finallyit’s less than epislon

diverge - AP to show there is some number for which the function is always positive and therefore, there is always a number greater

Let M > 0, By AP ∃ n ∈ Naturals such that N > max (4,M) If n >gt N then n^2 03n = n(n-3) > n(4-3) ≥ N > M this limit goes to infinity