tronter said:1. Let [itex] A [/itex] be a nonempty set of real numbers which is bounded below. Let [itex] -A [/itex] be the set of numbers [itex] -x [/itex], where [itex] x \in A [/itex]. Prove that [itex] \inf(A) = -\sup(-A) [/itex].
Intuitively this makes sense if you draw it on a number line. But I am not sure how to formally prove it.
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.
A proof of lower bound is a mathematical justification or evidence that shows that a given set of real numbers has a lower bound.
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.
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.
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.