I have an exam tomorrow and could not solve the question below. Could you please help me?
Regards
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 $b_m = \sup\{x_n : n \ge m\}$, $b = \lim b_m$. Prove that $b \le c^∗$. (Hint: use (a).)