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7.E: Intermediate and Extreme Values (Exercises)

Last updated

May 28, 2022
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- 7.4: The Supremum and the Extreme Value Theorem
- 8: Back to Power Series

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Q1

Mimic the definitions of an upper bound of a set and the least upper bound (supremum) of a set to give definitions for a lower bound of a set and the greatest lower bound (infimum) of a set.

Note: The infimum of a set $S$ is denoted by $\inf (S)$ .

Q2

Find the least upper bound (supremum) and greatest lower bound (infimum) of the following sets of real numbers, if they exist. (If one does not exist then say so.)

$S = \{\frac{1}{n} |n = 1,2,3,...\}$
$T = \{r|r \text{ is rational and }r^2 < 2\}$
$(-∞,0) ∪ (1,∞)$
$R = \{\frac{(-1)^n}{n} |n = 1,2,3,...\}$
$(2,3π] ∩ \mathbb{Q}$
The empty set $\varnothing$

Q3

Let $S ⊆ R$ and let $T = \{-x|x ∈ S\}$ .

Prove that $b$ is an upper bound of $S$ if and only if $-b$ is a lower bound of $T$ .
Prove that $b = \sup S$ if and only if $-b = \inf T$ .

Q1

Q2

Q3

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