University of Toronto – MAT137Y1 – LEC0501
Calculus!
Characterization of the sup/inf (slide 6)
Jean-Baptiste Campesato January 9
th, 2019
Recall the following definitions from the videos.
Definition 1. Let𝐴 ⊆ ℝand𝑈 ∈ ℝ. We say that𝑈is anupper boundof𝐴if
∀𝑥 ∈ 𝐴, 𝑥 ≤ 𝑈
Definition 2. Let𝐴 ⊆ ℝand𝐿 ∈ ℝ. We say that𝐿is alower boundof𝐴if
∀𝑥 ∈ 𝐴, 𝐿 ≤ 𝑥
Definition 3. We say that a subset𝐴 ⊆ ℝisbounded from aboveif it admits an upper bound.
Definition 4. We say that a subset𝐴 ⊆ ℝisbounded from belowif it admits a lower bound.
Definition 5. Let𝐴 ⊆ ℝand𝑆 ∈ ℝ.
We say that𝑆is thesupremum(orleast upper bound) of𝐴if 1. 𝑆is an upper bound of𝐴, and,
2. for all upper bounds𝑇 of𝐴,𝑆 ≤ 𝑇. Then we use the notation𝑆 =sup(𝐴).
Definition 6. Let𝐴 ⊆ ℝand𝐼 ∈ ℝ.
We say that𝐼is theinfimum(orgreatest lower bound) of𝐴if 1. 𝐼is a lower bound of𝐴, and,
2. for all lower bounds𝐽of𝐴,𝐽 ≤ 𝐼.
Then we use the notation𝐼 =inf(𝐴).
Remark 7. Notice that we talk aboutthesupremum of a set but aboutanupper bound of a set.
It is because, as seen during the lecture (slide 4), if a set admits a supremum then it is unique.
Beware, it is possible for a set to not have a supremum.
The real lineℝsatisfies two very fundamental properties.
Theorem 8(The least upper bound property).
If a non-empty subset ofℝis bounded from above then it admits a least upper bound (supremum).
Theorem 9(The greatest lower bound property).
If a non-empty subset ofℝis bounded from below then it admits a greatest lower bound (infimum).
Remark 10. As seen during the lecture (slide 5), the “non-empty” assumption is essential here!
2 Characterization of the sup/inf (slide 6)
We have seen the following characterizations of the supremum and of the infimum (slide 6).
These characterizations may be useful when writing proofs: do not hesitate to use them!
Proposition 11. Let𝐴 ⊆ ℝand𝑆 ∈ ℝ. Then 𝑆 =sup(𝐴) ⇔{
∀𝑥 ∈ 𝐴, 𝑥 ≤ 𝑆
∀𝜀 > 0, ∃𝑥 ∈ 𝐴, 𝑆 − 𝜀 < 𝑥 Proposition 12. Let𝐴 ⊆ ℝand𝐼 ∈ ℝ. Then
𝐼 =inf(𝐴) ⇔ {
∀𝑥 ∈ 𝐴, 𝐼 ≤ 𝑥
∀𝜀 > 0, ∃𝑥 ∈ 𝐴, 𝑥 < 𝐼 + 𝜀
We will only focus on the characterization of the supremum (that’s similar for the infimum).
Notice that the first line simply means that𝑆is an upper bound.
Then the second line of the characterization means that𝑆is the smallest one!
Indeed, for any𝜀 > 0, even a very very very small one,𝑆 − 𝜀 < 𝑆. So the fact that𝑆is the least upper bound means exactly that𝑆 − 𝜀isn’t an upper bound, or, equivalently, that there is at least one𝑥 ∈ 𝐴such that𝑆 − 𝜀 < 𝑥.
𝐴 ℝ
𝑆 𝑆 − 𝜀
𝜀
𝑥
Beware, for simplicity I represented𝐴as an interval in the above figure, but 𝐴may not be an interval!
Proof of proposition 11. Let𝐴 ⊆ ℝand𝑆 ∈ ℝ.
1. Proof of⇒.
Assume that𝑆 =sup(𝐴).
Then𝑆is a upper bound of𝐴so∀𝑥 ∈ 𝐴, 𝑥 ≤ 𝑆.
We know that if𝑇 is an upper bound of𝐴then𝑆 ≤ 𝑇.
So, by taking the contrapositive, if𝑇 < 𝑆then𝑇 isn’t an upper bound of𝐴.
Let𝜀 > 0. Since𝑆 − 𝜀 < 𝑆, we know that𝑆 − 𝜀is not an upper bound of𝐴, meaning that there exists𝑥 ∈ 𝐴such that𝑆 − 𝜀 < 𝑥.
2. Proof of⇐.
We assume that
{
∀𝑥 ∈ 𝐴, 𝑥 ≤ 𝑆
∀𝜀 > 0, ∃𝑥 ∈ 𝐴, 𝑆 − 𝜀 < 𝑥
The first part of the characterization ensures that𝑆is an upper bound of𝐴.
We still have to prove that if𝑇 is an upper bound of𝐴then𝑆 ≤ 𝑇. We will show the contrapositive: if𝑇 < 𝑆then𝑇 isn’t an upper bound.
Let𝑇 ∈ ℝ. Assume that𝑇 < 𝑆. Let𝜀 = 𝑆 − 𝑇 > 0. Then there exists𝑥 ∈ 𝐴such that 𝑆 − 𝜀 < 𝑥, i.e.𝑇 < 𝑥.
Hence𝑇 isn’t an upper bound.
■ Remark 13. If you prefer, you can write proofs by contradiction instead of using the contrapositive.
