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

- Page ID
- 9028

[ "article:topic", "Extreme Value Theorem", "authorname:eboman", "Supremum Value Theorem" ]

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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 \(-b = \inf T\).

### Contributor

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