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alexmahone
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Let $S$ and $T$ be non-empty subsets of $\mathbb{R}$, and suppose that for all $s\in S$ and $t\in T$, we have $s\le t$. Prove that $\sup S\le\inf T$.
Clearly $ \inf(T)$ & $\sup(S) $ exist. WHY?Alexmahone said:Let $S$ and $T$ be non-empty subsets of $\mathbb{R}$, and suppose that for all $s\in S$ and $t\in T$, we have $s\le t$. Prove that $\sup S\le\inf T$.
Plato said:Clearly $ \inf(T)$ & $\sup(S) $ exist. WHY?
$\forall s\in S$ we know $s\le\inf(T)~.$ WHY?
Well every $s\in S$ is a lower bound for $T$.Alexmahone said:I don't know. Why?
Plato said:Well every $s\in S$ is a lower bound for $T$.
Therefore $s\le\inf(T)$.
Yes. It is correct.Alexmahone said:So if $\sup S>\inf T$, there must be an $s\in S$ such that $s>\inf T$. (Otherwise, $\inf T$ is a smaller upper bound for $S$ than $\sup S$.) So we get a contradiction. (Is that correct?)
The supremum of a set is the smallest upper bound of the set, while the infimum is the largest lower bound of the set.
Supremum and infimum are used to determine the limits of a set or sequence. They are also important concepts in the field of calculus, where they are used to find maximum and minimum values of functions.
This phrase refers to a specific problem in which one is asked to find the supremum and infimum of a given set or sequence. These types of problems are common in mathematical analysis and can involve a variety of techniques and methods.
To solve an another supremum and infimum problem, you first need to understand the definition of supremum and infimum and how they are used in mathematical analysis. Then, you can apply different techniques such as using the definitions, properties, and theorems of supremum and infimum, as well as using calculus methods and mathematical reasoning to find the solutions.
Sure. For example, you may be asked to find the supremum and infimum of the set {1/n | n ∈ N}. In this case, the supremum is 1 and the infimum is 0, as 1 is the smallest upper bound and 0 is the largest lower bound of the given set.