Here is the proof of the statement on the title that has been given to me. I understand everything except the last line. Perhaps someone could enlighten me.
Let $c'$ be a cluster point of $\{x_n\}$ and $c=\lim \inf x_n > c'$. Then there exists a subsequence $\{x_{n_k}\}$ of $\{ x_{n} \}$ such that $\lim_{k\rightarrow\infty} x_{n_k}=c'$.
In other words, $\forall \ \epsilon>0, \ \exists \ N \in \mathbb{N}$ such that $ \forall \ k\geq N$ we have $\mid x_{n_k}-c'\mid <\epsilon$. Choose $\epsilon = \frac{c-c'}{2}$. Then there are infinitely many $ x_{n_k}$ such that $x_{n_k} \leq c'+\epsilon = c-\epsilon$. Hence $c=\lim \inf x_n=c-\epsilon$ which contradicts $c'<c$.
I do not understand how $c=c-\epsilon$ contradicts $c'<c$. The only contradiction I see is that $c=c-\epsilon$ is problematic since $\epsilon \neq 0$ but I do not see how this implies that $c$ has to be less than or equal to $c'$. What am I missing?