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## Homework Statement

Let A be a nonempty set of real numbers which is bounded below. Let -A be the set of all numbers -x, where x is in A. Prove that:

inf A = -Sup(-A)

## Homework Equations

What should I use as a starting statement? I understand that its true. It makes sense in my head but I can't get it out on paper. l believe this to be right, although I'm not sure if my assumptions are correct or if what I did is actually legal to do. This is from Rudin's Principlese of Mathematical Analysis - Page 22 - Problem 6

## The Attempt at a Solution

By definition --

inf A --> x <= y where y is in A

sup(-A) --> -x >= -y where y is in A

Thus --

-sup(-A) --> -(-x >= -y) --> x <= y

So we see --

-sup(-A) --> x <= y where y is in A

Therefore --

inf A = -sup(-A)