1.

Let S = {0.1 ,0.11 ,0.111 ,0.1111 ,0.11111 ,0.111111,...}.

a. Show that elements of S are all less than 1/9.

b. Given t < 1/9, describe how you can find s S that is bigger than t.

Completing these two steps would allow you to conclude that supS = 1/9.

2.

Let S, T be two non-empty sets of real numbers, both bounded. Let U = {u R : u = s t, s S, t T }. Prove that infU = infS sup T .

3.

Let S, T be two non-empty sets of real numbers, both bounded. Suppose that

For every element s S, there exists t T such that s < t.

For every element t T , there exists s S such that t < s. Prove that supS = supT .