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I am stuck in my one of the homework problems, the question is like the following:

Let $(x_n)$ be a bounded sequence, and let $c$ be the greatest cluster point of $(x_n)$:

(a) Prove that for every $\epsilon > 0 $ there is $N$ such that for $n > N$ we have $x_n < c + \epsilon.\;$ (Hint: use the Bolzano-Weierstrass theorem.)

(b) Let $b_m = \text{sup}\{x_n : n >=m\};\; b = \text{lim}\; b_m$. Prove that $b \le c.\;$ (Hint: use (a).)

Can anyone give me a hand please? Thanks

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    This is the verbatim reproduction of the text of a homework you were given to do. What did you try? Where are you stuck?2012-11-11
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    This is my homework and I am stuck. I am not sure how to use Bolzano Weierstrass thm here. Can you please help me do it?2012-11-11

1 Answers 1