I just read a proof that says 0 < a ==> 0 < a^-1.

How do I prove 0 < a < b ==> 0 < b^-1 < a^-1?

Also, the text did not give a definition for 0 < a < b.

Would it be correct to assume 0 < a and a < b <=> 0 < a < b, similarly for 0<= a and a<=b <=> 0 <= a <= b?