Supremum of a set

[Baron]

Let A and B be nonempty subsets of the interval [0,1].
Let C be the set defined by C = {a*b s.t a ∈ A and b ∈ B}.

If sup(A) = sup(B) = 1, prove that sup(C) = 1

My work:

Since sup(A) = sup(B) = 1, for any a ∈ A and b ∈ B, a ≤ 1 and b ≤ 1.

Hence, ab ≤ 1. Since ab ∈ C, 1 is an upper bound for C.

Let m∈ R (real number) such that m < 1. I'm having trouble showing that there is an a ∈ A and b ∈ B, such that m < ab < 1

 

[HallsofIvy]

Since m<1, \(\displaystyle \sqrt{m}< 1\) also. Since 1 is the LUB for both A and B, there exist a in A and b in B such that \(\displaystyle \sqrt{m}< a< 1\) and \(\displaystyle \sqrt{m}< b< 1\). What can you say about ab?