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.

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- September 22nd 2008, 07:33 PMSnooks02Boundedness and Convergence
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.

- September 22nd 2008, 07:35 PMJhevon
- September 22nd 2008, 08:00 PMThePerfectHacker
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.