Show if true or provide a counterexample if false If A and B

Show if true or provide a counterexample if false:

If A and B are two nonempty bounded sets of real numbers and l.u.b. A < l.u.b. B, then for each x in A, there exists a y in B such that x is less than y.

Solution

Since , A and B are non empty sets and sup(A) < sup(B)

Let sup(A) = a and sup(B) = b and a < b

Lets say A = { a₁ , a₂ , a₃ . . .a_n , a } ( arranged in ascending order )

and B = { b₁ , b₂ , b₃ . . . b_n , b } ( arranged in ascending order )

Let x = a₁ in A hence a₁ < a < b => a₁ < b

Let x = a₂ in A hence a₂ < a < b => a₂ < b

.. ..

Let x = a_n in A hence a_n < a < b => a_n < b

Therefore , there always exists an element x in A and y in B for which x y