2.4 Applications of the Supremum Property

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Proposition: Example 2.4.1 (a)

Let $S\ne \varnothing ,S\subseteq \mathbb{R}$ that is bounded above. $\forall a\in \mathbb{R}$: **$$ \sup(a + S) = a + \sup S

Proposition: Example 2.4.1 (b)

Let $A,B\subseteq \mathbb{R},A,B\ne \varnothing$. $a<b\forall a\in A,b\in B$, then **$$ \sup A \leq \inf B

Definition: Bounded function

Misalkan diberikan suatu fungsi $f:D\to \mathbb{R}$ dan range $f$, $f(D)=\{F(x):x\in D\}$

• Fungsi $f$ dikatakan terbatas di atas jika $f(D)$ terbatas di atas, yakni

  $$\exists B\in \mathbb{R}\ni f(x)\le B\forall x\in D$$

• Fungsi $f$ dikatakan terbatas di bawah jika $f(D)$ terbatas di bawah, yakni

  $$\exists L\in \mathbb{R}\ni B\le f(x)\forall x\in D$$

• Fungsi $f$ dikatakan terbatas jika $f$ terbatas di atas dan di bawah, yang ekivalen dengan

  $$\exists M\in \mathbb{R}\ni |f(x)|\le M\forall x\in D$$

Proposition: Example 2.4.2 (a)

Misalkan $f,g$ fungsi bernilai real dengan domain $D\subseteq \mathbb{R}$. Misalkan pula $f,g$ terbatas. Jika $f(x)\le g(x)$ untuk setiap $x\in D$ maka,

$$\underset{x\in D}{\sup }f(x)\le \underset{x\in D}{\sup }g(x).$$

Proposition: Example 2.4.2 (c)

Misalkan $f,g$ fungsi bernilai real dengan domain $D\subseteq \mathbb{R}$. Misalkan pula $f,g$ terbatas. Jika $f(x)\le g(y)$ untuk setiap $x,y\in D$ maka,

$$\underset{x\in D}{\sup }f(x)\le \underset{x\in D}{\sup }g(x).$$

2.4.3 Theorem: Archimedean property

If $x\in \mathbb{R}$, then there exists $n_x\in \mathbb{N}$ such that $x\le n_x$.

Other forms:

• $x\in \mathbb{R}\to \exists n_x\in \mathbb{N}\ni x\le n_x$.

• There is always a bigger natural number than any arbitrary real number.

For proofing, use:

• Proof by contradiction
• Completeness Property
• $\mathbb{N}$

2.4.4 Corollary: From 2.4.3

If $S:=\{\frac{1}{n}:n\in \mathbb{N}\}$, then $\inf S=0$.

$S:=\{\frac{1}{n}:n\in \mathbb{N}\}\to \inf S=0.$

2.4.5 Corollary

If $t>0$, then there exists $n_t\in \mathbb{N}$ such that $0<\frac{1}{n_t}<t$.

$t>0\to \exists n_t\in \mathbb{N}\ni 0<\frac{1}{n_t}<t.$

2.4.6 Corollary

If $y>0$, then there exists $n_y\in \mathbb{N}$ such that $n_y-1\le y\le n_y$.

$y>0\to \exists n_y\in \mathbb{N}\ni n_y-1\le y\le n_y.$

2.4.7 Theorem: Existence of irrational number

There exists a positive real number $x$ such that $x^2=2$.

$\exists x\in \mathbb{P}\ni x^2=2.$

2.4.8 Theorem: Density of rational numbers at R

If $x,y\in \mathbb{R}$ such that $x<y$, then there exists $r\in \mathbb{Q}$ such that $x<r<y$.

$x,y\in R,x<y\to \exists r\in \mathbb{Q}\ni x<r<y.$

2.4.9 Corollary: From 2.4.8

If $x,y\in \mathbb{R}$ such that $x<y$, then there exists $t\in {\mathbb{Q}}^c$ such that $x<t<y$.

$x,y\in R,x<y\to \exists t\in {\mathbb{Q}}^c\ni x<t<y.$