So I am working through Hardy's classic on my own and I need to get some feedback on how to proceed with the questions at the end of each section. Right now I am working on p. 18 example 4: Prove 0=-0.
Now Hardy gives the clue that all of the examples on p. 18 are consequences of the definitions that precede, which are thus:
" A section of the rational numbers, in which both classes exist and the lower class has no greatest member, is called a real number, or simply a number...(1)"
" It will be convenient to define at this stage the negative −α of a positive number α. If (a), (A) are the classes which constitute α, we can define another section of the rational numbers by putting all numbers −A in the lower class and all numbers −a in the upper...(2)"
Then to prove 0=-0 we would have to show, by the definitions above, that the upper and lower classes of 0,-0 are equal. It is intuitive that this must be true, but I am new to proofs and not sure how to tackle it. Maybe by contradiction? Don't give it away point me on the right path. Thanks