i am stuck in one of my 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).)
For part a), I tried to show the contrapositive, i supposed suppose there is an ϵ>0 such that infinitely many xn's satisfy xn≥c+ϵ, then this determines a subsequence that has a cluster point ≥c+ϵ. But I cannot completely explain this
For part b, I see that For all ϵ>0 we have bm < c+ϵ for almost all m (i.e., for m>N for a fixed N∈N). So i have to show that the sequence (bm) is bounded and monotonic. Thus, it has a limit, and so its limit satisfies b≤c+ϵ. But again, i could not fully explain it.
Can somebody give me a hand? Thanks