Decide if the following statements about suprema and infima

Decide if the following statements about suprema and infima
are true or false. Give a short proof for those that are true. For any that are
false, supply an example where the claim in question does not appear to hold.
(a) If A and B are nonempty, bounded, and satisfy A B, then sup A
sup B.
(b) If sup A < inf B for sets A and B, then there exists a c R satisfying
a<c<b for all a A and b B.
(c) If there exists a c R satisfying a<c<b for all a A and b B, then
sup A < inf B.

Solution

a) TRUE

if x is any upper bound for B, then x is also an upper bound for A because any a A will be present in B and hence a x. Upper bound for B will also be an upper bound for A but any upper bound of A is greater or equal then the least upper bound of A so sup(A) sup(B).

b) TRUE

If supA<infB then we can say that every element in set A is smaller then every element in set B so a<b for a belonging to A and b belonging to B and also least upper bound of A is less then inf B then in between these two numbers there will can be a real number c.

c) FALSE

because for example A=(1,2) and B=(2,3) then a<c<b is satisfied for a belonging to A and b belonging to B and value of c can be 2 but sup A= inf B =2