Alright, I need some guidance with the following problem. Any help would be appreciated.
Suppose and given any the set contains both poisitive and negative real numbers,show that .
After I talked to some people it sounds like there may be a typo???
Possibly in this problem? Either way I'm not really sure how to show that b=0.
Attempt #1 at proof:
Suppose where contains only negative real numbers and contains only positive real values. Let be a limit point of and . Then let be the function . Thus, the exists only if both exist and are equal.
Then, if both limits exist we have that, for some ,when , . Also, when , .
The only way that this can happen is when ?????
I think you confused stuff here big time: the positive and negative numbers are in the image of f , NOT in its domain!
We know that .
Suppose , and let us choose . By the above, we get that for any
point , for some ,or in other words: ...but this can't be since
in there's some negative number, whereas in all the numbers are positive...
Read the above slowly and with a pencil and a piece of paper by your side...draw diagrams if that helps you.
Tonio