Suppose that {x(n)} 1=n=infinity, is a sequence of real numbers with limit x, and suppose that a≤ x(n) ≤b, all n. Prove that a≤ x ≤b.
If is convergent and then - where .
This is what we will prove. Say we prove that if then . This is sufficient to complete the proof. Why? Because if with then define so and by above (here ).
Now let . And assume, by contradiction, . Then there is so that and so a contradiction.
Similary do it for by repeating the argument.