Let x be an upper bound for A R Prove that if x SolutionGive

Let x be an upper bound for A R. Prove that if x

Solution

Given that x is upper bound of A then by definition of upper bound every element of A is less than x

a) suppose that x<y since x is upper bound of A and x<y means every element of A is less than y means y is upper bound of A

b) suppose x is in A

Given that x is upper bound means every element of A is less than x and x is element of A itself hence it is the smallest upper bound of A

Therefore x is least upper bound of A

I.e x=Sup(A)