If $X$ is a non-compact space then prove that $C_b(X)$ is not separable, where $C_b(X)$ is space of all bounded continuous functions on $X$.
I was trying like this, but got stuck at middle: Take a sequence $S$ consisting of distinct points without having a limit point. Now for all subset $A$ of that sequence we can get a continuous function $f_A$ such that $f_A\rvert_{A}=1$ and vanishes at $S-A$. So $A\neq B$ implies $\lvert f_A-f_B \rvert=1$ and thus we'll get an uncountable collection of functions $\{f_I\}$ such that $\lvert f_i-f_j \rvert=1 \;\forall i,j \in I$. Now I'm stuck.
Now you have finished. Indeed you have found an uncountable number of disjoint balls. For example the balls of center $f$ and radius $1/2$. A dense subset should intersect all these disjoint balls, so there is at least one point in every ball. So the dense set must be uncountable.