# Proof of lower bound of a nonempty set of real numbers

• tronter
In summary, if A is a nonempty set of real numbers bounded below, and -A is the set of numbers -x where x ∈ A, then it can be proven that the infimum of A is equal to the negative of the supremum of -A. This can be seen intuitively by drawing it on a number line, but a formal proof can be established by showing that -sup(-A) is a lower bound of A and is the least lower bound.
tronter
1. Let $A$ be a nonempty set of real numbers which is bounded below. Let $-A$ be the set of numbers $-x$, where $x \in A$. Prove that $\inf(A) = -\sup(-A)$.

Intuitively this makes sense if you draw it on a number line. But I am not sure how to formally prove it.

tronter said:
1. Let $A$ be a nonempty set of real numbers which is bounded below. Let $-A$ be the set of numbers $-x$, where $x \in A$. Prove that $\inf(A) = -\sup(-A)$.

Intuitively this makes sense if you draw it on a number line. But I am not sure how to formally prove it.

If -x is in -A then -x<sup(-A)
this implies:
x>-sup(-A)

Therefor -sup(-A) is a lower bound of A.

Now establish it is the least lower bound (probably via contradiction).

## What is a lower bound?

A lower bound is a number that is greater than or equal to all the numbers in a given set. It provides a limit or boundary for the set.

## What is a proof of lower bound?

A proof of lower bound is a mathematical justification or evidence that shows that a given set of real numbers has a lower bound.

## Why is it important to prove the lower bound of a set of real numbers?

Proving the lower bound of a set of real numbers is important because it helps to establish the existence and limit of the set. It also provides a foundation for further mathematical analysis and calculations.

## What are the different methods used to prove the lower bound of a set of real numbers?

There are several methods used to prove the lower bound of a set of real numbers, including the method of contradiction, the method of mathematical induction, and the method of direct proof.

## Can a set of real numbers have more than one lower bound?

Yes, a set of real numbers can have more than one lower bound. In fact, a set can have infinitely many lower bounds, as long as they are all greater than or equal to all the numbers in the set.

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