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