Constructing a bounded function with no min/max values using a particular sequence of points Let $(X, d)$ be a noncompact metric space and let $\{x_{n}\}$ be a sequence of points without a convergent subsequence. Choose a sequence of positive numbers $\{\epsilon_{n}\}$ such that $B_{\epsilon_{n}}(x_{n}) \cap B_{\epsilon_{m}}(x_{m}) = \emptyset$, whenever $n \neq m$. Suppose we define real-valued function $f_{n}: B_{\epsilon_{n}}(x_{n}) \rightarrow \mathbb{R}$ by
$$f_{n}(x) = \frac{\epsilon_{n} - d(x,x_{n})}{c_{n} + d(x,x_{n})}$$ where $c_{n} > 0$.
Then define function $f(x) = f_{n}(x)$ whenever $x \in B_{\epsilon_{n}}(x_{n})$ and $0$ otherwise. 
I have been able to show that this function is continuous, and that $c_{n}$ can be chosen so that $f$ is unbounded. But how would I go about choosing $c_{n}$ so that the function is bounded, but does not achieve a minimum or maximum value?
 A: First I'd like to point out that $f : \cup_{n=1}^{\infty} B_{\epsilon_n}(x_n) \rightarrow \mathbb{R},$ NOT $f: X \rightarrow \mathbb{R}$. 
Now, since it is required that $B_{\epsilon_n}(x_n)$ are such that they are pairwise disjoint, it is sufficient to choose $c_n$ to make it so that for every $n, f_n$ does not achieve a maximum or minimum. The easiest way to do this requires assuming that $B_{\epsilon_n}(x_n)$ are open balls. Then letting $c_n = \epsilon_n$ should do the trick, because we now have that $0 < f_n < 1$, but $\inf{f_n} = 0$ and $\sup{f_n} = 1$. To see the supremum, take a sequence $\{x_k\} \rightarrow x_n$ and to see the supremum, let $y_n$ be a point on the boundary of the ball $B_{\epsilon_n}(x_n)$, then choose a sequence $\{x_k\} \rightarrow y_n$. Therefore, $0 < f < 1$ and $0 = \inf f, 1 = \sup f$.
Also note this can be immediately generalized to work for any sequence $\{c_n\}$ so that $\inf \{ \frac{\epsilon_n}{c_n} \} > 0$, and $\sup \{ \frac{\epsilon_n}{c_n} \} < \infty$, but then $0 < f < \sup \{ \frac{\epsilon_n}{c_n} \}$, with $\inf f = 0, \sup f = \sup \{ \frac{\epsilon_n}{c_n} \}$.
I think you can fill in any additional detail you desire from here.
