PS from real analysis 2 Thank youSolutionA B sup B is an u

Solution

A B , sup( B ) is an upper bound for A . Since sup( A ) is the least upper bound for A by definition, it must be less than or equal sup( B ) .

Proof By Contradiction :

Let A B but sup( A ) > sup( B )

sup( A ) = a and sup( B ) = b => a > b

Since , A B hence , sup( A ) B and sup( B ) B

=> sup( A ) = sup( B ) which contradicts the condition

=> sup( A ) sup( B )