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# 7.E: Intermediate and Extreme Values (Exercises)

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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.)

1. $$S = \{\frac{1}{n} |n = 1,2,3,...\}$$
2. $$T = \{r|r \text{ is rational and }r^2 < 2\}$$
3. $$(-∞,0) ∪ (1,∞)$$
4. $$R = \{\frac{(-1)^n}{n} |n = 1,2,3,...\}$$
5. $$(2,3π] ∩ \mathbb{Q}$$
6. The empty set $$\varnothing$$

## Q3

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

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

## Contributor

• Eugene Boman (Pennsylvania State University) and Robert Rogers (SUNY Fredonia)