The question asks:
Suppose that ab > 0. Show that if a < b, then 1/b < 1/a.
I know this is correct, but I don't know how to show it. I did this:
aa^-2 < ba^-2
1/a < ba^-2
(b^-2)/a < (a^-2)/b
Hoping that I could find a way to swap the inequality sign around whilst creating the fractions required. But I can't seem to do it.
Any assistance would be great!
One more thing worth mentioning since "show that" means we're dealing with formal proofs here.
These lines are connected implicitly by "if and only if" as in
1/b<1/a
1/b-1/a<0
{a-b}/ab<0
For the last line, we don't need to think about whether it's "if and only if" because we only need to go in one direction
1/b<1/a
1/b-1/a<0
{a-b}/ab<0
I write this mainly so that nobody gets confused, thinking the proof is not valid because we assumed what we wanted to prove.