6.3 Supremum and infimum[29M]
Let’s first review the characterizations of the supremum and infimum in \(ℝ\), as seen in Sec. ?? (or in Chap. 1 Sect. 5 in the notes [ 2 ] ). Let \(A⊆ ℝ\) be a non empty set.
[08T] Let \(A ⊆ ℝ\) be not be empty. Recall that the supremum, or least upper bound, of a set \(A\) is the minimum of majorants; We will indicate it with the usual writing \(\sup A\). If \(A\) is bounded above then \(\sup A\) is a real number; otherwise, by convention, it is set to \(\sup A=+∞\).
[208](Solved on 2022-11-24) Let therefore \(A⊆ℝ\) be not empty, let \(l ∈ ℝ ∪ \{ +∞\} \); you can easily demonstrate the following properties:
|
\(\sup A ≤ l\) |
\(∀ x∈ A,x≤ l\) |
|
\(\sup A {\gt} l\) |
\(∃ x∈ A,x{\gt} l\) |
|
\(\sup A {\lt} l\) |
\(∃ h{\lt}l , ∀ x∈ A,x≤ h\) |
|
\(\sup A ≥ l\) |
\(∀ h{\lt}l, ∃ x∈ A,x{\gt} h\) |
the first and third derive from the definition of supremum,
1
the second and fourth by negation; in the third we can conclude equivalently that \(x{\lt}h\), and in the fourth that \(x≥ h\).
If \(l≠ +∞\) then also we can also write (replacing \(h=l-\varepsilon \))
|
\(\sup A {\lt} l\) |
\(∃ \varepsilon {\gt}0 , ∀ x∈ A,x≤ l-\varepsilon \) |
|
\(\sup A ≥ l\) |
\(∀ \varepsilon {\gt}0, ∃ x∈ A,x{\gt} l-\varepsilon \) |
[209] Similarly, given \(A⊆ℝ\) not empty, the greatest lower boundary, or infimum, of \(A\) is the maximum of minorants; we will indicate it with the usual writing \(\inf A\). If \(A\) is bounded below then \(\inf A\) is a real number; otherwise, by convention, we set \(\inf A=-∞\).
[0B5](Proposed on 2022-11-24) Note that if we replace \(A\) with
and \(l\) with \( -l\), we switch from the definitions of \(\sup \) to those of \(\inf \) (and vice versa).
[20B] Let \(A ⊆ ℝ\) not empty, let \(l∈ℝ∪\{ -∞\} \); the following properties apply:
|
\(\inf A ≥ l\) |
\(∀ x∈ A,x≥ l\) |
|
\(\inf A {\lt} l\) |
\(∃ x∈ A,x{\lt}l\) |
|
\(\inf A {\gt} l\) |
\(∃ h{\gt}l , ∀ x∈ A,x≥ h\) |
|
\(\inf A ≤ l\) |
\(∀ h{\gt}l, ∃ x∈ A,x{\lt} h\) |
If \(l≠ -∞\) then also we write (substituting \(h=l+\varepsilon \))
|
\(\inf A {\gt} l\) |
\(∃ \varepsilon {\gt}0 , ∀ x∈ A,x≥ l+\varepsilon \) |
|
\(\inf A ≤ l\) |
\(∀ \varepsilon {\gt}0, ∃ x∈ A,x≤ l+\varepsilon \) |
[20M](Proposed on 2022-11-24) Having fixed \(A⊆ℝ\) not empty, then \(\inf A\) is the only number \(𝛼∈ℝ∪\{ -∞\} \) which satisfies these two properties
[20H] (Solved on 2022-11-24) Given \(J\) an index set (not empty), let \(a_ n∈ℝ\) for \(n∈ J\). The supremum and infimum are defined as
where \(A=\{ a_ n: n∈ J\} \) is the image of the sequence.
Given \(D\) not empty, let \(f:D→ℝ\) be a function. The supremum and infimum are defined as
where \(A=\{ f(x): x∈ D\} \) is the image of the function.
Exercises
Let \(I,J\) be generic non-empty sets. See definitions in Sec. 6.3
- E188
[0B6] (Solved on 2022-11-24) Let \(a_ n\) be a real-valued sequence, for \(n∈ I\) a set of indexes; let \(r{\gt}0,t∈ℝ,𝜌{\lt}0\); show that
\[ \sup _{n∈ I}(a_ n+t)=t+\sup _{n∈ I}a_ n~ ~ ,~ ~ \sup _{n∈ I}(r a_ n)=r \sup _{n∈ I}a_ n~ ~ ,~ ~ \sup _{n∈ I}(𝜌 a_ n)= 𝜌 \inf _{n∈ I}a_ n~ ~ . \]Hidden solution: [UNACCESSIBLE UUID ’22W’]
- E188
[0B7] (Solved on 2022-11-24) Let \(a_{n,m}\) be a real sequence with two indices \(n∈ I,m∈ J\), show that
\[ \sup _{n∈ I,m∈ J}a_{n,m} = \sup _{n∈ I} \Big(\sup _{m∈ J} a_{n,m} \Big)~ ~ . \]Hidden solution: [UNACCESSIBLE UUID ’0B8’]
- E188
[0B9] Prerequisites:2,1.(Solved on 2022-11-24) Let \(a_ n,b_ n\) be real sequences, for \(n∈ I\), show that
\[ \sup _{n,m∈ I}(a_ n+b_ m) = (\sup _{n∈ I} a_ n) + (\sup _{n∈ I} b_ n)~ ~ , \]but
\[ \sup _{n∈ I}(a_ n+b_ n)≤ (\sup _{n∈ I} a_ n) + (\sup _{n∈ I} b_ n)~ ~ ; \]find a case where inequality is strict. Hidden solution: [UNACCESSIBLE UUID ’0BB’]
- E188
[0BC] Prerequisites:2. Let \(A,B⊆ ℝ\) and let
\[ A ⊕ B=\{ x + y : x∈ A, y∈ B\} \]the Minkowski sum 2 of the two sets: show that
\[ \sup (A ⊕ B)=(\sup A) + (\sup B)~ ~ . \]Hidden solution: [UNACCESSIBLE UUID ’0BD’]
- E188
[0BF]Let \(I_ n⊆ ℝ\) (for \(n∈ℕ\)) be closed and bounded non-empty intervals, such that \(I_{n+1}⊆ I_ n\): show that \(⋂_{n=0}^∞ I_ n\) is not empty.
This result is known as Cantor’s intersection theorem [ 36 ] . It is valid in more general contexts, see 9 and 3.
If we replace \(ℝ\) with \(ℚ\) and assume that \(I_ n⊆ ℚ\), is the result still valid?
- E188
[20P](Solved on 2022-11-24) Study the equivalences in proposition 182 for the case in which \(\sup A=+∞\): What do the formulas on the right say?
- E188
[20J]Rewrite the properties of the clause 186 for the cases seen in 188.
- E188
[20Y](Proposed on 2022-12) Calculate supremum and infimum of the following sets (where \(n,m\) are integers).
\begin{eqnarray*} \left\{ \frac{mn}{m^ 2+n^ 2}: n,m≥ 1 \right\} \quad ,\quad \left\{ \frac{mn}{m+n}: n,m≥ 1 \right\} \\ \left\{ 2^ n+2^ m : n,m∈ℕ \right\} \quad ,\quad \left\{ 2^ n+2^ m: n,m∈ℤ \right\} \\ \left\{ \frac{m^ 2-2} n : n,m∈ℤ,n≠ 0 \right\} \quad ,\quad \left\{ \frac{m+1}{m^ 2} : m∈ℤ,m≠ 0 \right\} \end{eqnarray*}