One more thing worth mentioning since "show that" means we're dealing with formal proofs here.

1/b<1/a

1/b-1/a<0

{a-b}/ab<0

These lines are connected implicitly by "if and only if" as in

1/b<1/a

\(\displaystyle \displaystyle \iff\) 1/b-1/a<0

\(\displaystyle \displaystyle \iff\) {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

\(\displaystyle \displaystyle \iff\) 1/b-1/a<0

\(\displaystyle \displaystyle \iff\) {a-b}/ab<0

\(\displaystyle \displaystyle \Longleftarrow ab > 0 \land a < b\)

I write this mainly so that nobody gets confused, thinking the proof is not valid because we assumed what we wanted to prove.